Dimension formulas for modular form spaces of rational weights, the classification of eta-quotient characters and an extension of Martin’s theorem

Xiao-Jie Zhu School of Mathematical Sciences
Key Laboratory of MEA (Ministry of Education) & Shanghai Key Laboratory of PMMP
East China Normal University
500 Dongchuan Road, 200241
Shanghai, P. R. China [email protected] https://orcid.org/0000-0002-6733-0755

Abstract.

We give an explicit formula for dimensions of spaces of rational-weight modular forms whose multiplier systems are induced by eta-quotients of fractional exponents. As the first application, we give series expressions of Fourier coefficients of the $n$ -th root of certain infinite $q$ -products. As the second application, we extend Yves Martin’s list of multiplicative holomorphic eta-quotients of integral weights by first extending the meaning of multiplicativity, then identifying one-dimensional spaces, and finally applying Wohlfahrt’s extension of Hecke operators. A table containing $2277$ of such eta-quotients is presented. As a related result, we completely classify the multiplier systems induced by eta-quotients of integral exponents. For instance, there are totally $384$ such multiplier systems on $\Gamma_{0}(4)$ for any fixed weight. We also provide SageMath programs on checking the theorems and generating the tables.

Key words and phrases:

modular form, dimension formula, rational weight, Dedekind eta function, Hecke operator, multiplicative eta-quotient

2020 Mathematics Subject Classification:

Primary 11F12; Secondary 11F20, 11F25, 11F30, 11L05, 30F10

This work is supported in part by Science and Technology Commission of Shanghai Municipality (No. 22DZ2229014).

1. Introduction

1.1. Dimension formulas

A holomorphic function $f$ defined on the upper half plane $\mathfrak{H}$ is called a modular form if it satisfies two conditions: first $f\left(\frac{a\tau+b}{c\tau+d}\right)=(c\tau+d)^{k}\chi\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right)f(\tau),\,\tau\in\mathfrak{H}$ for all $\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right)$ in some discrete matrix group $G$ , $k$ being an integer or half integer, called the weight, and $\chi$ being a multiplier system of $G$ ; second $f$ is holomorphic at all cusps of $G$ , the meaning of which will be clarified in Definition 2.1. Let the space of all modular forms with given $G$ , $k$ and $\chi$ be denoted by $M_{k}(G,\chi)$ .

Modular forms, spaces of modular forms and their variants play important roles in mathematics. In the study of these topics, having an explicit formula for the dimension of the complex vector space $M_{k}(G,\chi)$ is crucial to some applications. For instance, $\dim_{\mathbb{C}}M_{k}(G,\chi)=1$ would imply that any function $f$ in $M_{k}(G,\chi)$ is an eigenfunction for a family of operators called the Hecke operators and hence we obtain nontrivial relations among the Fourier coefficients of $f$ .

A natural and elegant way for computing $\dim_{\mathbb{C}}M_{k}(G,\chi)$ is to use the Riemann-Roch theorem and the Riemann-Hurwitz formula. Petersson [1, p. 194] applied this method and gave a dimension formula which he called the generalized Riemann-Roch theorem. Petersson’s formula concerns arbitrary Fuchsian group $G$ of the first kind, arbitrary complex weight $k$ and arbitrary multiplier system $\chi$ while this formula is not so explicit in the sense one can not directly compute the dimension using elementary operations and numerical information about $G$ and $\chi$ . To our best knowledge, Shimura [2, Section 2.4] gave explicit formulas for $G$ arbitrary, $k$ being an integer and $\chi$ trivial first. Another way for computing $\dim_{\mathbb{C}}M_{k}(G,\chi)$ is to use the Eichler-Selberg trace formula for Hecke operators; c.f. [3, 4] for the case $G$ being the full modular group, $\chi$ trivial and [5] for the case $G=\Gamma_{0}(N)$ and $\chi$ induced by a Dirichlet character modulo $N$ , both dealing with integral weights. See also [6]. For the half-integral weights, Cohen and Oesterlé [7] gave an explicit dimension formula for the case $G=\Gamma_{0}(4N)$ , $\chi$ being the product of the multiplier system of the theta series $\sum_{n\in\mathbb{Z}}\mathrm{e}^{2\uppi\mathrm{i}n^{2}\tau}$ and that of a Dirichlet character modulo $4N$ . Since then, there appear many excellent works on computing $\dim_{\mathbb{C}}M_{k}(G,\chi)$ for certain special $G$ or special $\chi$ , or computing the dimensions of certain subspaces; c.f. [8], [9], [10] and [11].

Modular forms of rational weights are less attractive to mathematicians than those of integral or half-integral weights. We know little about them. For instance, it seems no Hecke theory has been attempted. (Maybe there is no such theory.) However, there actually exist many such forms and more importantly, there are interesting applications, e.g., to noncongruence modular forms. Explicit dimension formulas for spaces of rational weights have been obtained by Ibukiyama [12, Lemma 1.7], [13, p. 5] in some special cases, using essentially the method of Petersson [1], but no proof is given.

As our main result, we give explicit formulas for $\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(N),\chi)$ where $k$ is any rational number, $\chi$ is the multiplier system of any level $N$ eta-quotient $\prod_{n\mid N}\eta(n\tau)^{r_{n}}$ of fractional exponents $r_{n}$ . See Definition 2.1 for the exact meaning of $M_{k}(\Gamma_{0}(N),\chi)$ and (6) for that of $\chi$ . The proof is based on Petersson’s method [1] and we provide the full details. The following theorem is a special case of Theorem 4.2, which is our main theorem.

A special case of the Main Theorem.

If $k>2-\frac{6}{m}\varepsilon_{2}-\frac{8}{m}\varepsilon_{3}-\frac{12}{m}\sum_{c\mid N}\phi(c,N/c)\cdot\left(1-\left\{\frac{x_{c}}{24}\right\}\right)$ , then

(1)

\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(N),\chi)=\frac{k-1}{12}m+\frac{1}{4}\varepsilon_{2}+\frac{1}{3}\varepsilon_{3}+\sum_{c\mid N}\phi(c,N/c)\cdot\left(\frac{1}{2}-\left\{\frac{x_{c}}{24}\right\}\right),

where $m$ , $\varepsilon_{2}$ and $\varepsilon_{3}$ , depending on $N$ , are given in (9), (10) and (11) respectively and

k=\frac{1}{2}\sum_{n\mid N}r_{n},\quad x_{c}=\sum_{n\mid N}\frac{N}{(N,c^{2})}\cdot\frac{(n,c)^{2}}{n}r_{n}\quad\text{ for }c\mid N.

Note that the notation $\phi(c,N/c)$ refers to the Euler totient $\phi$ of the greatest common divisor of $c$ and $N/c$ .

In particular, if $k\geq 2$ , then (1) always holds. For any rational $k$ with $0<k<2$ , there are also infinitely many pairs $(N,\chi)$ such that $\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(N),\chi)$ can be computed using (1). See Section 7 for complete lists of such pairs $(N,\chi)$ for $k=1/2,\,1,\,3/2$ and $r_{n}$ being integers. The dimensions for $k=1/2$ and $k=3/2$ in the case $G=\Gamma_{0}(4N)$ and $\chi$ is the product of the multiplier system of the theta series $\sum_{n\in\mathbb{Z}}\mathrm{e}^{2\uppi\mathrm{i}n^{2}\tau}$ and that of a Dirichlet character modulo $4N$ are previously known: Serre and Starks [14] gave explicit bases of the spaces of weight $1/2$ ; Cohen and Oesterlé [7] established relations between spaces of weight $1/2$ and those of weight $3/2$ . The important and interesting problem of computing dimensions in weight $1$ is difficult and the most complete result about this was obtained by Deligne and Serre [15]. Using their theory, one can compute $\dim_{\mathbb{C}}M_{1}(\Gamma_{0}(N),\chi)$ for any $N\in\mathbb{Z}_{\geq 1}$ and $\chi$ induced by any Dirichlet character modulo $N$ although there is no explicit formula.

The difference between our formulas for $\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(N),\chi)$ with $k\in\frac{1}{2}\mathbb{Z}$ and others’ is that we deal with those $\chi$ induced by eta-quotients while the existing formulas are about those $\chi$ induced by Dirichlet characters or the multiplier system of $\sum_{n\in\mathbb{Z}}\mathrm{e}^{2\uppi\mathrm{i}n^{2}\tau}$ . Moreover, when $k\in\mathbb{Q}\setminus\frac{1}{2}\mathbb{Z}$ , our formulas include the one in [12, Lemma 1.7] and [13, p. 5] as special cases.

After preparing, stating and proving the main theorem and establishing some related tools (Proposition 5.1, Theorem 6.8, Tables 1, LABEL:table:wt1 and Section 7.3), we will give mainly two applications, both of which are based on identifying one-dimensional spaces. The first application concerns rational-weight modular forms and the second concerns ordinary eta-quotients whose exponents $r_{n}$ are all integers.

1.2. Application I

We give a series expression of the Fourier coefficients of the $n$ -th root of certain infinite $q$ -products. See Corollary 8.8 for the details. To avoid overlapping with the main text, here we give a randomly chosen example which is different from Examples 8.9 and 8.10.

Proposition 1.1.

We have

\frac{\sqrt[3]{\prod_{n\in\mathbb{Z}_{\geq 1}}(1-q^{n})}\cdot\sqrt{\prod_{n\in\mathbb{Z}_{\geq 1}}(1-q^{2n})^{29}}}{\sqrt[3]{\prod_{n\in\mathbb{Z}_{\geq 1}}(1-q^{4n})^{22}}}=1+\mathrm{e}^{-\frac{15\uppi\mathrm{i}}{8}}\frac{(2\uppi)^{\frac{15}{4}}}{\Gamma(\frac{15}{4})}\\ \cdot\sum_{n\in\mathbb{Z}_{\geq 1}}n^{\frac{11}{4}}\left(\sum_{c\in\mathbb{Z}_{\geq 1}}\frac{1}{(4c)^{\frac{15}{4}}}\sum_{\begin{subarray}{c}{0\leq d<4c}\\ {(d,4c)=1}\end{subarray}}\mathfrak{e}\left(\frac{dn}{4c}-\frac{1}{24}P(4c,d;\frac{29}{2},-\frac{22}{3})\right)\right)\cdot q^{n},

where $q=\mathrm{e}^{2\uppi\mathrm{i}\tau}$ throughout the paper, $\tau\in\mathfrak{H}$ and

	$\displaystyle P(4c,d;\frac{29}{2},-\frac{22}{3})=-88(-d,c)+174(-d,2c)+4s(-d,4c)-\frac{45}{2},$
	$\displaystyle s(-d,c)=\sum_{r=1}^{c-1}\frac{r}{c}\left(\left\{\frac{-dr}{c}\right\}-\frac{1}{2}\right).$

Corollary 8.8 contains infinitely many such identities of level $4$ , extending a previous result of the author [16, Theorem 7.1] which concerns levels $2$ and $3$ . For the outline of the proof, say of Proposition 1.1, let the infinite product be $f$ , which is an eta-quotient of fractional exponents. Then $f\in M_{15/4}(\Gamma_{0}(4),\chi)$ where $\chi$ is the multiplier system of $f$ . There is as well an Eisenstein series $g$ in $M_{15/4}(\Gamma_{0}(4),\chi)$ ; c.f. Definition 8.3. Now we apply Theorem 4.2 and find that $\dim_{\mathbb{C}}M_{15/4}(\Gamma_{0}(4),\chi)=1$ . Hence $f$ and $g$ are proportional. The identity thus follows by figuring out the Fourier coefficients of $g$ .

1.3. Application II

This is the major application. In 1996, Martin [17] obtained the complete list¹¹1There is a misprint in [17, p. 4853]. The entry $2^{-1}\cdot 4^{4}\cdot 6^{-1}\cdot 8^{-1}\cdot 12^{4}\cdot 24$ should be corrected to $2^{-1}\cdot 4^{4}\cdot 6^{-1}\cdot 8^{-1}\cdot 12^{4}\cdot 24^{-1}$ . of integral-weight holomorphic eta-quotients $f$ (with integral exponents) satisfying that

(a)

the multiplier system of $f$ is induced by some Dirichlet character,
(b)

if we write $f=\sum_{n\in\mathbb{Z}_{\geq 0}}c_{f}(n)q^{n}$ , then $c_{f}(n_{1})c_{f}(n_{2})=c_{f}(n_{1}n_{2})c_{f}(1)$ for any coprime positive integers.

We will discard condition (a) and seek for arbitrary holomorphic eta-quotients (with integral exponents) satisfying a multiplicativity property similar to the one in condition (b). We will also give many interesting new identities involving Fourier coefficients of these eta-quotients. Let us first present some randomly chosen examples. For the basic knowledge about Dedekind eta function $\eta(\tau)$ and eta-quotients, see the second half of Section 2.

Example.

Let $f(\tau)=\eta(\tau)^{-7}\eta(2\tau)^{17}\eta(4\tau)^{-3}=\sum_{n\in\frac{5}{8}+\mathbb{Z}_{\geq 0}}c_{f}(n)q^{n}$ . Let $l_{1},l_{2}$ be coprime odd square-free integers. Then

c_{f}\left(\frac{5l_{1}^{2}}{8}\right)c_{f}\left(\frac{5l_{2}^{2}}{8}\right)=c_{f}\left(\frac{5l_{1}^{2}l_{2}^{2}}{8}\right).

See Proposition 9.33 for details and the proof.

Example.

Let $f(\tau)=\eta(3\tau)^{2}\eta(9\tau)^{-1}\eta(27\tau)=\sum_{n\in\mathbb{Z}_{\geq 1}}c_{f}(n)q^{n}$ . Then $c_{f}(l_{1})c_{f}(l_{2})=c_{f}(l_{1}l_{2})$ whenever $l_{1},l_{2}\equiv 1\bmod{3}$ are square-free positive integers with $(l_{1},l_{2})=1$ . See Example 9.29 for details.

Example.

Let $f(\tau)=\eta(\tau)^{1}\eta(2\tau)^{-1}\eta(3\tau)^{-1}\eta(4\tau)^{1}\eta(6\tau)^{4}\eta(12\tau)^{-2}=\sum_{n\in\mathbb{Z}_{\geq 0}}c_{f}(n)q^{n}$ . Then

c_{f}(l)=-\sum_{a\mid l}\genfrac{(}{)}{}{}{a}{6}=-\prod_{p\mid l}\left(1+\genfrac{(}{)}{}{}{p}{6}\right)

where $1\leq l\equiv 1\bmod{24}$ is square-free. Note that $\genfrac{(}{)}{}{}{a}{6}$ refers to the Kronecker-Jacobi symbol. See Example 9.23 for details.

Martin [17] proved his results using Hecke operators in the sense of [18] while we prove our results using Wohlfahrt’s extension of Hecke operators [19]. See also [20, Section 3] for a theory of the double coset version of these operators. These operators are denoted by $T_{l}$ , where $l$ runs through a multiplicative submonoid $L$ of the positive integers; c.f. Corollary 9.13. As a prerequisite, we give an explicit formula for the action on Fourier coefficients; c.f. Theorem 9.15. Then we aim to find holomorphic eta-quotient $f$ such that $T_{l}f=c_{l}\cdot f$ with $c_{l}\in\mathbb{C}$ . At this point Theorem 4.2 enters into play. If $f$ , whose level is $N$ and character is $\chi$ , lies in a one-dimensional space $M_{k}(\Gamma_{0}(N),\chi)$ or $S_{k}(\Gamma_{0}(N),\chi)$ , the subspace of cusp forms, then we will have $T_{l}f=c_{l}\cdot f$ for all $l\in L$ . ( $L$ depends on $f$ .) This fact, which is stated in Theorem 9.19, is the main theorem of this application.

It seems that there are more than 10000 eta-quotients that are Hecke eigenforms (in the sense of (83)) for infinitely many $l$ . We list 2277 of them in Table LABEL:table:admissibleTypeI. They are what we call admissible eta-quotients of type I with levels

N=1,2,3,4,5,6,7,8,9,10,11,13,14,15,17,19,21,27.

For the meaning of admissible eta-quotients of type I, see the beginning part of Section 9. For other levels, either there is no admissible eta-quotient of type I, or there are too many so it is not appropriate to list them in the paper. One can find the SageMath code that generates admissible eta-quotients of type I or II with any given level in Appendix A.

Besides Theorem 9.19, another important result is Theorem 9.24, which says that

c_{l_{1}}\cdot c_{l_{2}}=c_{l_{1}l_{2}},\quad l_{1},l_{2}\in L,\quad(l_{1},l_{2})=1.

Here $c_{l}=T_{l}f/f$ . This is the genuine multiplicativity property that is satisfied by all admissible eta-quotients and reduces to the relation $c_{f}(n_{1})c_{f}(n_{2})=c_{f}(n_{1}n_{2})c_{f}(1)$ for ordinary multiplicative eta-quotients in [17].

All identities in the above three examples are immediate consequences of Theorems 9.19 and 9.24.

1.4. Other results, structure of the paper and notations

As a related result, we give the complete classification of linear characters that are induced by eta-quotients $\prod_{n\mid N}\eta(n\tau)^{r_{n}}$ with $r_{n}\in\mathbb{Z}$ on the double cover of $\Gamma_{0}(N)$ . See Corollary 6.9 for the conclusion and Examples 6.10, 6.11, 6.12, 6.13, 6.14 for examples. For instance, for each $k\in\frac{1}{2}\mathbb{Z}$ , there are totally $384$ linear characters of the double cover of $\Gamma_{0}(4)$ that are induced by $\eta(\tau)^{r_{1}}\eta(2\tau)^{r_{2}}\eta(4\tau)^{r_{n}}$ of weight $k$ . As a comparison, there are only two characters of $\Gamma_{0}(4)$ that are induced by Dirichlet characters modulo $4$ and these two characters are also contained in the above $384$ characters with any fixed $k\in\mathbb{Z}$ .

The structure of the paper is as follows. We review some elements of rational-weight modular forms in Section 2. The second half of this section contains elements of Dedekind eta function and eta-quotients of fractional exponents. In Section 3 we associate a divisor on certain compact Riemann surface with any rational-weight meromorphic modular form and in addition, we give a detailed proof of the valence formula in the case of rational weights. In Section 4 we state and prove the main theorem via Petersson’s method [1]. The following three sections contain some tools which are needed by the following two applications: Section 5 contains a formula that relates the orders at cusps and the exponents (or the character) of an eta-quotient. This formula is due to Bhattacharya (cf. [21, eq. (5.13)]). Here we give more details on the proof. In Section 6 we classify all linear characters that are induced by eta-quotients $\prod_{n\mid N}\eta(n\tau)^{r_{n}}$ with $r_{n}\in\mathbb{Z}$ on the double cover of $\Gamma_{0}(N)$ , as is described in the last paragraph. Section 7, which is subdivided into three subsections, contains tables of dimensions of modular form spaces of weight $1/2$ , $1$ and $3/2$ that can be computed by our main theorem. In Sections 8 and 9 we carry out the first and second applications, which has been described in Sections 1.2 and 1.3, respectively. Section 10 contains some comments, open problems and conjectures. Finally, many formulas in this paper have been verified and many tables are generated by SageMath [22] programs. Appendix A contains the usage of the code.

We collect some notations. For a set $A$ , the symbols $\lvert A\rvert$ and $\mathop{\mathrm{\#}}A$ both denote its cardinality. The notation $\mathbb{Q}^{A}$ denotes the $\mathbb{Q}$ -vector space of functions from $A$ into $\mathbb{Q}$ . A multiset $\{x_{i}\colon i\in I\}$ is defined to be the ordinary set $\{(y,n)\colon n=\mathop{\mathrm{\#}}\{i\in I\colon x_{i}=y\},\,n\neq 0\}$ . Its underlying (ordinary) set is still denoted by $\{x_{i}\colon i\in I\}$ . If $f\colon X\rightarrow Y$ is a function, then $f|_{A}$ is its restriction to $A$ where $A$ is a subset of $X$ . If $f,g$ are functions then $f\circ g$ is their composition: $f\circ g(x)=f(g(x))$ . For a family of sets $A_{i}$ , $i\in I$ , the notation $\bigsqcup_{i\in I}A_{i}$ denotes the union and only when the family $A_{i}$ are disjoint can one use this notation.

For a group $G$ , $\langle X\rangle$ denotes the subgroup generated by $X$ where $X$ is a subset of $G$ . If $H$ is a subgroup of $G$ then $[G\colon H]=\mathop{\mathrm{\#}}G/H=\mathop{\mathrm{\#}}H\backslash G$ is the index. If $G$ acts on the left (right resp.) on a set $X$ , then $G\backslash X$ ( $X/G$ resp.) denotes the set of orbits $G\cdot x$ ( $x\cdot G$ resp.) where $x\in X$ . If X is itself a group, $G$ is a subgroup of $X$ and the action is the group operation, then $G\backslash X$ ( $X/G$ resp.) is called the left coset space (right coset space resp.) For $x\in X$ , $G_{x}$ denotes the stabilizer, that is, the subgroup of $g\in G$ such that $gx=x$ (or $xg=x$ ). A character $\chi$ of $G$ is a complex linear character, that is, a group homomorphism from $G$ to $\mathbb{C}^{\times}$ , the multiplicative group of nonzero complex numbers. If $\lvert\chi(g)\rvert=1$ for all $g\in G$ , then $\chi$ is called unitary. If $G^{\prime}$ is another group and $\pi\colon G\rightarrow G^{\prime}$ is a group surjection, then we say the character $\chi$ descends to a character on $G^{\prime}$ if there is a character $\chi^{\prime}$ of $G^{\prime}$ such that $\chi=\chi^{\prime}\circ\pi$ .

The symbol $\phi$ refers to the Euler totient function. For integers $a,b,\dots$ , the notation $(a,b,\dots)$ denotes the greatest common divisor and $\phi(a,b,\dots)$ denotes the value of $\phi$ at $(a,b,\dots)$ . Let $N\in\mathbb{Z}_{\geq 1}$ ; the summation range of $\sum_{n\mid N}$ is implicitly understood to be the positive divisors of $N$ . Let $p$ be a prime and $\alpha\in\mathbb{Z}_{\geq 0}$ ; then $p^{\alpha}\parallel N$ means $p^{\alpha}\mid N$ but $p^{\alpha+1}\nmid N$ . The $p$ -adic exponential valuation, $v_{p}(N)$ , is the largest $\alpha$ such that $p^{\alpha}\parallel N$ . An empty sum is understood to be $0$ and an empty product be $1$ . For a real number $x$ , the notation $[x]$ means the largest integer not exceeding $x$ and $\{x\}=x-[x]$ . The functions $\Gamma(s)$ and $\zeta(s)$ are the usual Euler Gamma function and Riemann zeta function respectively.

Let $m,n\in\mathbb{Z}$ ; the Kronecker-Jacobi symbol $\genfrac{(}{)}{}{}{m}{n}$ is defined as follows:

•

$\genfrac{(}{)}{}{}{m}{p}$ is the usual Legendre symbol if $p$ is an odd prime.
•

$\genfrac{(}{)}{}{}{m}{2}$ equals $0$ if $2\mid m$ , and equals $(-1)^{(m^{2}-1)/8}$ if $2\nmid m$ .
•

$\genfrac{(}{)}{}{}{m}{-1}$ equals $1$ if $m\geq 0$ , and equals $-1$ otherwise.
•

$\genfrac{(}{)}{}{}{m}{1}=1$ by convention.
•

$\genfrac{(}{)}{}{}{m}{n}$ is defined to make it a complete multiplicative function of $n\in\mathbb{Z}-\{0\}$ .
•

$\genfrac{(}{)}{}{}{m}{0}=0$ if $m\neq\pm 1$ , and $\genfrac{(}{)}{}{}{\pm 1}{0}=1$ .

We shall freely use the following properties, especially in the proof of Theorem 9.15.

•

$\genfrac{(}{)}{}{}{m}{n_{1}n_{2}}=\genfrac{(}{)}{}{}{m}{n_{1}}\genfrac{(}{)}{}{}{m}{n_{2}}$ . When $m=-1$ , it is required that $n_{1},n_{2}\neq 0$ .
•

$\genfrac{(}{)}{}{}{m_{1}m_{2}}{n}=\genfrac{(}{)}{}{}{m_{1}}{n}\genfrac{(}{)}{}{}{m_{2}}{n}$ . When $n=-1$ , it is required that either $m_{1},m_{2}\neq 0$ , or one of $m_{1}$ , $m_{2}$ is $0$ and the other is nonnegative.
•

$\genfrac{(}{)}{}{}{m}{n}\genfrac{(}{)}{}{}{n}{m}=\varepsilon(m,n)\cdot(-1)^{\frac{n-1}{2}\cdot\frac{m-1}{2}}$ where $m,n$ are coprime odd integers, $\varepsilon(m,n)=-1$ if $n,m<0$ and $\varepsilon(m,n)=1$ otherwise.
•

$\genfrac{(}{)}{}{}{-1}{n}=(-1)^{\frac{n-1}{2}}$ and $\genfrac{(}{)}{}{}{2}{n}=(-1)^{\frac{n^{2}-1}{8}}$ where $n$ is odd.
•

The function $n\mapsto\genfrac{(}{)}{}{}{m}{n}$ is $\lvert m\rvert$ -periodic if $m\equiv 0,1\bmod{4}$ ; it is $\lvert 4m\rvert$ -periodic if $m\equiv 2\bmod{4}$ .
•

The function $m\mapsto\genfrac{(}{)}{}{}{m}{n}$ is $n$ -periodic if $n$ is odd and positive.

For the proofs, see [23, Section 2.2.2].

2. Modular forms of rational weight

In this section, we review some elements of the theory of modular forms of rational weight (cf. [16, 13, 24, 25, 12, 1]). The concept of modular forms of rational weight is a special case of that of generalized modular forms which was initiated by Knopp and Mason [25]. The multiplier system of a generalized modular form is not required to be unitary or to have finite order, while in this paper, the multiplier systems of all modular forms that occur are unitary and have finite orders. In addition, it seems that the modular forms of rational weights that occur in this paper can only be regarded as modular forms on non-congruence subgroups of $\mathrm{SL}_{2}(\mathbb{Z})$ with trivial character. As a comparison, Freitag and Hill [26] have recently constructed modular forms of weight $1/3$ on $\mathrm{SU}(2,1)$ whose levels are certain congruence subgroups of $\mathrm{SU}(2,1)$ . Furthermore, see [27, Section 2] for harmonic weak Maass forms of real weight and their relations with weakly holomorphic modular forms.

First some notations. Let $\mathrm{GL}_{2}^{+}(\mathbb{R})$ be the group of all $2\times 2$ real matrices with positive determinants and $\mathrm{SL}_{2}(\mathbb{R})$ be the subgroup of $\mathrm{GL}_{2}^{+}(\mathbb{R})$ whose elements have determinant $1$ . Let $D$ be a positive integer. Then we define the $D$ -cover of $\mathrm{GL}_{2}^{+}(\mathbb{R})$ by

\widetilde{\mathrm{GL}_{2}^{+}(\mathbb{R})^{D}}=\left\{\left(\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right),{\varepsilon}\left({c}^{\prime}\tau+{d}^{\prime}\right)^{\frac{1}{D}}\right)\colon\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right)\in\mathrm{GL}_{2}^{+}(\mathbb{R}),\,\varepsilon^{D}=1\right\},

where $\left(\begin{smallmatrix}{a^{\prime}}&{b^{\prime}}\\ {c^{\prime}}&{d^{\prime}}\end{smallmatrix}\right)$ is the matrix in $\mathrm{SL}_{2}(\mathbb{R})$ proportional to $\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right)$ and $\left(c^{\prime}\tau+d^{\prime}\right)^{\frac{1}{D}}$ means a function of $\tau\in\mathfrak{H}=\{z\in\mathbb{C}\colon\Im(z)>0\}$ . The notation $\left(\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right),{\varepsilon}\left({c}^{\prime}\tau+{d}^{\prime}\right)^{\frac{1}{D}}\right)$ is sometimes abbreviated to $\left(\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right),{\varepsilon}\right)$ when $D$ is understood. The composition is given by

\left(\left(\begin{smallmatrix}{a_{1}}&{b_{1}}\\ {c_{1}}&{d_{1}}\end{smallmatrix}\right),{\varepsilon_{1}}\left({c_{1}}^{\prime}\tau+{d_{1}}^{\prime}\right)^{\frac{1}{D}}\right)\cdot\left(\left(\begin{smallmatrix}{a_{2}}&{b_{2}}\\ {c_{2}}&{d_{2}}\end{smallmatrix}\right),{\varepsilon_{2}}\left({c_{2}}^{\prime}\tau+{d_{2}}^{\prime}\right)^{\frac{1}{D}}\right)\\ =\left(\left(\begin{smallmatrix}a_{1}&b_{1}\\ c_{1}&d_{1}\end{smallmatrix}\right)\left(\begin{smallmatrix}a_{2}&b_{2}\\ c_{2}&d_{2}\end{smallmatrix}\right),\varepsilon_{1}\varepsilon_{2}\left(c_{1}^{\prime}\frac{a_{2}^{\prime}\tau+b_{2}^{\prime}}{c_{2}^{\prime}\tau+d_{2}^{\prime}}+d_{1}^{\prime}\right)^{\frac{1}{D}}\left(c_{2}^{\prime}\tau+d_{2}^{\prime}\right)^{\frac{1}{D}}\right).

It can be verified straightforwardly that $\widetilde{\mathrm{GL}_{2}^{+}(\mathbb{R})^{D}}$ with this composition is a group. One should note that different choices of holomorphic branches of $\left(c^{\prime}\tau+d^{\prime}\right)^{\frac{1}{D}}$ lead to different $D$ -covers. We choose the following branch throughout:

z^{r}=\exp\left(r\log z\right),\quad-\uppi<\Im(\log z)\leq\uppi.

It is possible to make $\widetilde{\mathrm{GL}_{2}^{+}(\mathbb{R})^{D}}$ a topological covering of $\mathrm{GL}_{2}^{+}(\mathbb{R})$ and hence $\widetilde{\mathrm{GL}_{2}^{+}(\mathbb{R})^{D}}$ becomes a Lie group whose smooth structure is transposed from $\mathrm{GL}_{2}^{+}(\mathbb{R})$ using the above covering map (cf. [28]). The motivation for using $D$ -covers is that when dealing with modular forms of weight $k\in\frac{1}{D}\mathbb{Z}$ , the multiplier systems become group characters on $D$ -covers.

Let $G$ be a subgroup of $\mathrm{GL}_{2}^{+}(\mathbb{R})$ ; then by $\widetilde{G^{D}}$ we understand the preimage of $G$ under the natural projection $\widetilde{\mathrm{GL}_{2}^{+}(\mathbb{R})^{D}}\rightarrow\mathrm{GL}_{2}^{+}(\mathbb{R})$ , $\left(\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right),{\varepsilon}\right)\mapsto\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right)$ . For $\gamma\in\mathrm{GL}_{2}^{+}(\mathbb{R})$ , let $\widetilde{\gamma}$ denote $(\gamma,1)\in\widetilde{\mathrm{GL}_{2}^{+}(\mathbb{R})^{D}}$ . (The cover index $D$ should be inferred from the context.) In addition, we set $\overline{G}=\left\{g\cdot\{\pm\left(\begin{smallmatrix}{1}&{0}\\ {0}&{1}\end{smallmatrix}\right)\}\colon g\in G\right\}$ , which is a subgroup of $\overline{\mathrm{GL}_{2}^{+}(\mathbb{R})}=\mathrm{GL}_{2}^{+}(\mathbb{R})/\{\pm\left(\begin{smallmatrix}{1}&{0}\\ {0}&{1}\end{smallmatrix}\right)\}$ .

The group $\mathrm{SL}_{2}(\mathbb{Z})$ is the set of $2\times 2$ integral matrices of determinant $1$ and is known as the full modular group. We also need the congruence subgroup

\Gamma_{0}(N)=\left\{\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right)\in\mathrm{SL}_{2}(\mathbb{Z})\colon c\equiv 0\bmod N\right\},

where $N$ is a positive integer called the level. It is well known that the index $[\mathrm{SL}_{2}(\mathbb{Z})\colon\Gamma_{0}(N)]=N\prod_{p\mid N}(1+1/p)$ where $p$ denotes a prime (cf. [29, Coro. 6.2.13]). The matrices $\left(\begin{smallmatrix}{1}&{0}\\ {0}&{1}\end{smallmatrix}\right)$ , $\left(\begin{smallmatrix}{1}&{1}\\ {0}&{1}\end{smallmatrix}\right)$ , $\left(\begin{smallmatrix}{0}&{-1}\\ {1}&{0}\end{smallmatrix}\right)$ and $\left(\begin{smallmatrix}{N}&{0}\\ {0}&{1}\end{smallmatrix}\right)$ are denoted by $I$ , $T$ , $S$ , $B_{N}$ respectively.

Let $\mathscr{M}(\mathfrak{H})$ denote the field of meromorphic functions on the upper half plane $\mathfrak{H}$ and $k\in\frac{1}{D}\mathbb{Z}$ . Define the slash operator of weight $k$ by

f|_{k}\left(\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right),{\varepsilon}\right)(\tau)=\left(\varepsilon\sqrt[D]{c^{\prime}\tau+d^{\prime}}\right)^{-Dk}f\left(\frac{a\tau+b}{c\tau+d}\right),

where $f\in\mathscr{M}(\mathfrak{H})$ , $\left(\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right),{\varepsilon}\right)\in\widetilde{\mathrm{GL}_{2}^{+}(\mathbb{R})^{D}}$ and $\tau\in\mathfrak{H}$ . It is immediate that $\widetilde{\mathrm{GL}_{2}^{+}(\mathbb{R})^{D}}$ acts on $\mathscr{M}(\mathfrak{H})$ on the right via the slash operator of weight $k$ . Let $G$ be a subgroup of $\mathrm{SL}_{2}(\mathbb{Z})$ and $\chi$ be a character²²2A character always means a unitary linear character, that is, a group homomorphism to the group of complex numbers of absolute value $1$ . on $\widetilde{G^{D}}$ . We say $f$ transforms like a modular form of weight $k$ and with multiplier system (or with character) $\chi$ if $f|_{k}\gamma=\chi(\gamma)f$ for any $\gamma\in\widetilde{G^{D}}$ . If this is the case, and $[\mathrm{SL}_{2}(\mathbb{Z})\colon G]<+\infty$ and $\chi(\gamma)$ has finite order for any $\gamma\in\widetilde{G^{D}}$ , then for $\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right)\in\mathrm{SL}_{2}(\mathbb{Z})$ we have

f|_{k}\widetilde{\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right)}(\tau)=\sum_{n\in m^{-1}\mathbb{Z}}c_{n}q^{n},

where $q=\mathfrak{e}\left(\tau\right)=\exp(2\uppi\mathrm{i}\tau)$ and $m\in\mathbb{Z}_{\geq 1}$ provided that there is $Y_{0}>0$ (depending on $\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right)$ ) such that $f|_{k}\widetilde{\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right)}$ has no poles on $\mathfrak{H}_{Y_{0}}=\{\tau\in\mathfrak{H}\colon\Im(\tau)>Y_{0}\}$ . Moreover, the series converges normally on $\mathfrak{H}_{Y_{0}}$ . To see this, note that there exists $m\in\mathbb{Z}_{\geq 1}$ such that $\chi\left(\widetilde{\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right)}\widetilde{T}^{m}\widetilde{\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right)}^{-1}\right)=1$ and hence $f|_{k}\widetilde{\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right)}(\tau+m)=f|_{k}\widetilde{\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right)}(\tau)$ . Therefore, the desired expansion follows from Fourier’s theorem or Laurent’s theorem. Define the order $\mathop{\mathrm{ord}}_{\mathrm{i}\infty}(f|_{k}(\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right),\varepsilon))$ to be the least $n$ such that $c_{n}\neq 0$ . If the expansion does not exist (that is, $f|_{k}(\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right),\varepsilon)$ has a nonisolated singularity at infinity) then the order is undefined; if the expansion holds but for any $n_{0}\in\mathbb{Z}$ there exists $n<n_{0}$ such that $c_{n}\neq 0$ then define $\mathop{\mathrm{ord}}_{\mathrm{i}\infty}(f|_{k}(\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right),\varepsilon))=-\infty$ ; if $f|_{k}(\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right),\varepsilon)$ is identically zero then define $\mathop{\mathrm{ord}}_{\mathrm{i}\infty}(f|_{k}(\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right),\varepsilon))=+\infty$ .

Definition 2.1.

Let $D\in\mathbb{Z}_{\geq 1}$ and $k\in\frac{1}{D}\mathbb{Z}$ . Let $G$ be a subgroup of $\mathrm{SL}_{2}(\mathbb{Z})$ of finite index and $\chi\colon\widetilde{G^{D}}\rightarrow\mathbb{C}^{\times}$ be a character of finite order³³3If $\chi$ is not of finite order as considered by Knopp and Mason [25], this definition also makes sense. (that is, $\chi(\gamma)$ has finite order for any $\gamma\in\widetilde{G^{D}}$ ). Let $f\in\mathscr{M}(\mathfrak{H})$ . Then we say $f$ is a meromorphic modular form of weight $k$ for $G$ with character $\chi$ if $f|_{k}\gamma=\chi(\gamma)f$ for any $\gamma\in\widetilde{G^{D}}$ and $\mathop{\mathrm{ord}}_{\mathrm{i}\infty}(f|_{k}\gamma)>-\infty$ for any $\gamma\in\widetilde{\mathrm{SL}_{2}(\mathbb{Z})^{D}}$ . Suppose this is the case. Then we say $f$ is a weakly holomorphic modular form if it has no poles on $\mathfrak{H}$ ; it is a modular function if the weight $k=0$ ; it is a modular form (cusp form respectively) if it is weakly holomorphic and $\mathop{\mathrm{ord}}_{\mathrm{i}\infty}(f|_{k}\gamma)\geq 0$ ( $>0$ respectively) for any $\gamma\in\widetilde{\mathrm{SL}_{2}(\mathbb{Z})^{D}}$ . The vector spaces (over $\mathbb{C}$ ) of modular forms and of cusp forms are denoted by $M_{k}(G,\chi)$ and $S_{k}(G,\chi)$ respectively. When $\chi$ is trivial (which means $k\in 2\mathbb{Z}$ and $\chi$ always takes the value $1$ ), we let $M_{k}(G)=M_{k}(G,\chi)$ and $S_{k}(G)=S_{k}(G,\chi)$ .

Note that the cover index $D$ can be recovered from the domain of $\chi$ and $M_{k}(G,\chi)$ is also denoted by $M_{k}(\widetilde{G^{D}},\chi)$ when needed.

Remark 2.2.

Suppose there exists a nonzero meromorphic modular form $f$ of weight $k$ for $G$ with character $\chi$ . Then we have the following facts:

•

$\chi\left(I,\mathfrak{e}\left(1/D\right)\right)=\mathfrak{e}\left(-k\right)$ .
•

If $-I\in G$ , then $\chi\left(\widetilde{-I}\right)=\mathfrak{e}\left(-k/2\right)$ .
•

If $D^{\prime}$ is another positive integer with $k\in\frac{1}{D^{\prime}}\mathbb{Z}$ and $D^{\prime}\mid D$ , then $\chi\colon\widetilde{G^{D}}\rightarrow\mathbb{C}^{\times}$ descends to a character on $\widetilde{G^{D^{\prime}}}$ , i.e., $\chi$ can be factored as $\chi=\chi_{1}p_{D,D^{\prime}}$ where $p_{D,D^{\prime}}\colon\widetilde{G^{D}}\rightarrow\widetilde{G^{D^{\prime}}}$ is the natural projection $(\gamma,\varepsilon)\mapsto(\gamma,\varepsilon^{D/D^{\prime}})$ and $\chi_{1}$ is a character on $\widetilde{G^{D^{\prime}}}$ (which is unique).

The proof is straightforward by considering slash operators acting on $f$ of which we omit the details. Motivated by the first two facts, we call a character $\chi$ on $\widetilde{G^{D}}$ with the properties $\chi\left(I,\mathfrak{e}\left(1/D\right)\right)=\mathfrak{e}\left(-k\right)$ and $\chi\left(\widetilde{-I}\right)=\mathfrak{e}\left(-k/2\right)$ if $-I\in G$ a multiplier system for $G$ of weight $k$ of cover index $D$ . Note that if $-I\in G$ , then the second property implies the first one. Finally, note that the third fact still holds when we require that $\chi$ is a multiplier system for $G$ of weight $k$ of cover index $D$ even if we can not find a nonzero form $f$ .

Remark 2.3.

If $f$ is a meromorphic modular form of weight $k$ for $G$ with character $\chi$ , and $g$ is a meromorphic modular form of weight $k^{\prime}$ for $G$ with character $\chi^{\prime}$ with $k,k^{\prime}\in\frac{1}{D}\mathbb{Z}$ , then it is immediate that $f\cdot g$ is a meromorphic modular form of weight $k+k^{\prime}$ for $G$ with character $\chi\cdot\chi^{\prime}$ . Moreover, if $f$ and $g$ are both modular forms (cusp forms respectively), then so is $f\cdot g$ .

One of the aims of this paper is to give explicit formulas concerning $\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(N),\chi)$ and $\dim_{\mathbb{C}}S_{k}(\Gamma_{0}(N),\chi)$ where $k\in\mathbb{Q}$ and $\chi$ is a character on $\widetilde{\Gamma_{0}(N)^{D}}$ of some kind which we explain now.

We need the Dedekind eta function and its logarithm, which are defined by

	$\displaystyle\eta(\tau)$	$\displaystyle=q^{1/24}\prod_{n\in\mathbb{Z}_{\geq 1}}\left(1-q^{n}\right),\quad\tau\in\mathfrak{H},$
(2)		$\displaystyle\log\eta(\tau)$	$\displaystyle=\log\eta(\mathrm{i})+\int_{\mathrm{i}}^{\tau}\frac{\eta^{\prime}(z)}{\eta(z)}\,\mathrm{d}z,$

where $\log\eta(\mathrm{i})$ is the real logarithm. The transformation equations of $\log\eta$ under $\mathrm{SL}_{2}(\mathbb{Z})$ are obtained by Dedekind (cf. [30, Equation (12), Section 3.4]):

(3)

\log\eta\left(\frac{a\tau+b}{c\tau+d}\right)-\log\eta(\tau)=2\uppi\mathrm{i}\left(\frac{a+d}{24c}+\frac{1}{2}s(-d,c)-\frac{1}{8}\right)+\frac{1}{2}\log(c\tau+d),

where $\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right)\in\mathrm{SL}_{2}(\mathbb{Z})$ with $c>0$ and $s(-d,c)$ is the Dedekind sum

(4)

s(h,k)=\sum_{r\bmod k}\Big{(}\Big{(}\frac{r}{k}\Big{)}\Big{)}\cdot\Big{(}\Big{(}\frac{hr}{k}\Big{)}\Big{)},\quad h\in\mathbb{Z},\,k\in\mathbb{Z}_{\geq 1}.

In the above definition, $((x))=x-[x]-1/2$ if $x\in\mathbb{R}\setminus\mathbb{Z}$ and $((x))=0$ if $x\in\mathbb{Z}$ . Set

(5)		$\displaystyle\Psi\colon\mathrm{SL}_{2}(\mathbb{Z})$	$\displaystyle\rightarrow\mathbb{Z}$
	$\displaystyle\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right)$	$\displaystyle\mapsto\begin{dcases}\frac{a+d}{c}+12s(-d,c)-3,&c>0;\\ \frac{a+d}{c}+12s(d,-c)+3,&c<0;\\ b,&c=0,a>0;\\ -b-6,&c=0,a<0.\end{dcases}$

Fix a positive integer $D$ . Then the transformation equations of $\eta$ follow from that of $\log\eta$ and can be expressed as $\eta|_{1/2}(\gamma,\varepsilon)=\chi_{\eta}(\gamma,\varepsilon)\eta$ , where $\chi_{\eta}$ is the multiplier system for $\mathrm{SL}_{2}(\mathbb{Z})$ of weight $1/2$ of cover index $2D$ defined by

\chi_{\eta}(\gamma,\varepsilon)=\varepsilon^{-D}\mathfrak{e}\left(\frac{\Psi(\gamma)}{24}\right),\quad(\gamma,\varepsilon)\in\widetilde{\mathrm{SL}_{2}(\mathbb{Z})^{2D}}.

The fact that $\Psi(\mathrm{SL}_{2}(\mathbb{Z}))\subseteq\mathbb{Z}$ (when we give (5) this is tacitly assumed) follows from the above formula with $D=1$ , the fact $\mathrm{SL}_{2}(\mathbb{Z})$ is generated by $T$ and $S$ and the fact $\chi_{\eta}(\widetilde{T})^{24}=\chi_{\eta}(\widetilde{S})^{24}=1$ .

Let $N,D\in\mathbb{Z}_{\geq 1}$ . By an eta-quotient of level $N$ and cover index $D$ we understand a product $f(\tau)=\prod_{0<n\mid N}\eta(n\tau)^{r_{n}}$ with $r_{n}\in\mathbb{Q}$ such that $D\cdot r_{n}\in 2\mathbb{Z}$ for any $n\mid N$ . Thus an eta-quotient of cover index $2$ is just an ordinary eta-quotient whose exponents are integers. The fractional powers are defined by $\eta(n\tau)^{r_{n}}=\exp\left(r_{n}\log\eta(n\tau)\right)$ (cf. (2)). Since $\eta$ has no poles on $\mathfrak{H}$ , the eta-quotient $f$ is a weakly holomorphic modular form of weight $k=\frac{1}{2}\sum_{n\mid N}r_{n}$ for $\widetilde{\Gamma_{0}(N)^{D}}$ with character

(6)		$\displaystyle\chi\colon\widetilde{\Gamma_{0}(N)^{D}}$	$\displaystyle\rightarrow\mathbb{C}^{\times}$
	$\displaystyle\left(\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right),{\varepsilon}\right)$	$\displaystyle\mapsto\varepsilon^{-Dk}\mathfrak{e}\left(\frac{1}{24}\sum_{n\mid N}r_{n}\Psi\left(\begin{smallmatrix}{a}&{bn}\\ {c/n}&{d}\end{smallmatrix}\right)\right).$

The above formula can be proved using (3), Remark 2.3 and the fact $\eta(n\tau)=n^{-1/4}\eta|_{1/2}\widetilde{B_{n}}(\tau)$ . Note that $\eta$ has no zeros on $\mathfrak{H}$ according to its infinite product expansion. Therefore any eta-quotient has no zeros or poles on $\mathfrak{H}$ . For the order at infinity, we have (cf. [16, Lemma 4.2])

(7)

\mathop{\mathrm{ord}}\nolimits_{\mathrm{i}\infty}(f|_{k}\widetilde{\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right)})=\frac{1}{24}\sum_{n\mid N}\frac{(n,c)^{2}}{n}r_{n},\quad\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right)\in\mathrm{SL}_{2}(\mathbb{Z}).

We recommend the reader to see [31, Section 2] or [21, Section 5] for more details on ordinary eta-quotients. If (7) are always nonnegative for all $\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right)\in\mathrm{SL}_{2}(\mathbb{Z})$ , we say $f$ is a holomorphic eta-quotient.

We will investigate the dimensions $\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(N),\chi)$ and $\dim_{\mathbb{C}}S_{k}(\Gamma_{0}(N),\chi)$ where $k\in\frac{1}{D}\mathbb{Z}$ and $\chi$ is the character of an eta-quotient of weight $k$ , level $N$ and cover index $D$ in the next three sections.

Remark 2.4.

The reader should be warned that when dealing with half-integral weights, our notation may differ from other authors’, e.g. from the notation encountered in [32] and [14]. Let $\psi$ be a Dirichlet character modulo $4N$ and let $2k\in\mathbb{Z}_{\geq 1}$ . The space $M_{0}(4N,k,\psi)$ in Serre and Stark’s notation is the same thing as $M_{k}(\Gamma_{0}(4N),\chi_{1}\psi_{1})$ in our notation where $\chi_{1}$ is the character of $\eta(\tau)^{-4k}\eta(2\tau)^{10k}\eta(4\tau)^{-4k}$ and $\psi_{1}$ is the character that maps $\left(\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right),{\varepsilon}\right)$ to $\psi(d)$ . The character $\chi_{1}$ is hidden in many authors’ notation of spaces and it sometimes appears in their definitions of slash operators or modular transformations or automorphic factors.

3. Divisors of modular forms

Our method relies on the classical Riemann-Roch theorem, so we recall some elements here. By a compact Riemann surface $X$ , we understand a compact connected Hausdorff topological space equipped with a maximal atlas $\{(U_{i},\phi_{i})\}$ (i.e., each $U_{i}$ is open in $X$ and $\phi_{i}$ is a homeomorphism from $U_{i}$ onto an open set in $\mathbb{C}$ ) such that when $(U_{i},\phi_{i})$ and $(U_{j},\phi_{j})$ overlap, the transition map $\phi_{j}\circ\phi_{i}^{-1}\colon\phi_{i}(U_{i}\cap U_{j})\rightarrow\phi_{j}(U_{i}\cap U_{j})$ is holomorphic. It can be shown that $X$ is second-countable (cf. [33, p. 88]). Therefore, according to the well known classification of compact connected topological surfaces (cf. eg. [34, Theorem 10.22]) and the fact that Riemann surfaces are oriented, $X$ is homeomorphic to the $2$ -sphere $\mathbb{S}^{2}$ or to the connected sum of $g$ copies of the real projective plane $\mathbb{P}^{2}$ with $g\in\mathbb{Z}_{\geq 1}$ . The number $g$ is called the genus of $X$ . (If $X$ is a sphere, then its genus is defined to be $0$ .)

A meromorphic differential $\omega$ on $X$ is by definition a holomorphic differential on $X\setminus S$ where $S$ is a finite subset of $X$ such that if $(U,\phi\colon U\rightarrow\phi(U)\subseteq\mathbb{C})$ is any chart of $X$ , then $\omega$ is of the form $f(z)\,\mathrm{d}z$ on this chart where $f(z)$ is a meromorphic function on $\phi(U)$ whose poles are exactly $\phi(U\cap S)$ . The basic feature of a meromorphic differential is the transformation equation between different charts: if $(V,\psi)$ is another chart that overlaps $(U,\phi)$ and $\omega=g(w)\,\mathrm{d}w$ on $(V,\psi)$ , then we have

f(z)=g(\psi\circ\phi^{-1}(z))\frac{\,\mathrm{d}\psi\circ\phi^{-1}(z)}{\,\mathrm{d}z},\quad z\in\phi(U\cap V).

The set of all meromorphic differentials on $X$ is denoted by $\mathscr{K}(X)$ which is obviously a complex vector space. Let $\mathscr{M}(X)$ denote the complex vector space of all meromorphic functions on $X$ as in Section 2. If there exists a nonzero $\omega_{0}\in\mathscr{K}(X)$ , then the map $\mathscr{M}(X)\rightarrow\mathscr{K}(X)$ that sends $f$ to $f\cdot\omega_{0}$ is a $\mathbb{C}$ -linear isomorphism. Such $\omega_{0}$ actually exists according to a fundamental result in the theory of Riemann surfaces (cf. [33, Theorem 1.10, Chapter IV]). As a consequence, nonzero meromorphic functions always exist on any compact Riemann surface.

By a ( $\mathbb{Q}$ -valued) divisor on $X$ , we understand a function $X\rightarrow\mathbb{Q}$ with finite support. We often write a divisor $D$ as a formal sum $\sum_{x\in X}d_{x}\cdot(x)$ where $d_{x}=D(x)$ . The term “divisor” traditionally refers to a $\mathbb{Z}$ -valued divisor but in this paper to a $\mathbb{Q}$ -valued divisor. The sets of all $\mathbb{Z}$ -valued divisors and of all $\mathbb{Q}$ -valued divisors are denoted by $\mathscr{D}(X)$ and $\mathscr{D}_{\mathbb{Q}}(X)$ respectively. Equipped with the pointwise addition, they are abelian groups. We define the degree of $D\in\mathscr{D}_{\mathbb{Q}}(X)$ by $\deg(D)=\sum_{x\in X}D(x)$ , which is well-defined since the sum is actually a finite sum. Moreover, define the floor function as follows: for $D\in\mathscr{D}_{\mathbb{Q}}(X)$ , set $[D]=\sum_{x\in X}[d_{x}]\cdot(x)$ , where $[d_{x}]$ is the greatest integer that does not exceed $d_{x}$ . Finally, for two divisors $D_{1}$ and $D_{2}$ , we say $D_{1}\geq D_{2}$ if $D_{1}(x)\geq D_{2}(x)$ far any $x\in X$ , and we say $D_{1}>D_{2}$ if $D_{1}\geq D_{2}$ but $D_{1}\neq D_{2}$ .

Generally speaking, with any meromorphic section of a holomorphic line bundle over $X$ one can associate a $\mathbb{Z}$ -valued divisor. Specifically, if $f\in\mathscr{M}(X)$ , then define $\mathop{\mathrm{div}}(f)=\sum_{x\in X}d_{x}\cdot(x)$ with $d_{x}$ being the least exponent in the Laurent expansion of $f$ at $x$ in any chart. For $\omega\in\mathscr{K}(X)$ we define $\mathop{\mathrm{div}}(\omega)$ in a similar manner.

The compact Riemann surfaces we need are the modular curves $X_{G}$ where $G$ is a finite index subgroup of $\mathrm{SL}_{2}(\mathbb{Z})$ which we describe now. Let $\mathfrak{H}^{*}=\mathfrak{H}\cup\mathbb{P}^{1}(\mathbb{Q})$ where $\mathbb{P}^{1}(\mathbb{Q})$ is the projective line over $\mathbb{Q}$ which can be identified with $\mathbb{Q}$ together with a point $\mathrm{i}\infty=1/0$ . The group $\mathrm{SL}_{2}(\mathbb{Z})$ acts on $\mathbb{P}^{1}(\mathbb{Q})$ on the left via $\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right)\cdot(p/q)=\frac{ap+bq}{cp+dq}$ . We endow $\mathfrak{H}^{*}$ with the topology generated by the usual topology of $\mathfrak{H}$ and the sets

\{\gamma\tau\colon\tau\in\mathfrak{H}_{Y_{0}}\cup\{\mathrm{i}\infty\}\},\quad\gamma\in\mathrm{SL}_{2}(\mathbb{Z}),\,Y_{0}>0,

where $\mathfrak{H}_{Y_{0}}=\{\tau\in\mathfrak{H}\colon\Im(\tau)>Y_{0}\}$ . It is immediate that $\mathfrak{H}^{*}$ with this topology is a Hausdorff space⁴⁴4The space $\mathfrak{H}^{*}$ is not locally compact and hence can not be a topological manifold., $\mathfrak{H}$ is an open subset of $\mathfrak{H}^{*}$ and $G$ acts on $\mathfrak{H}^{*}$ (on the left) by homeomorphisms. Let $G\backslash\mathfrak{H}^{*}$ be the orbit space (endowed with the quotient topology) and $p\colon\mathfrak{H}^{*}\rightarrow G\backslash\mathfrak{H}^{*}$ be the quotient map. It can be shown that $G\backslash\mathfrak{H}^{*}$ is Hausdorff (cf. eg. [35, Lemma 1.7.7]) and when $[\mathrm{SL}_{2}(\mathbb{Z})\colon G]<+\infty$ $G\backslash\mathfrak{H}^{*}$ is compact (cf. eg. [33, Proposition 14.6, Chapter IV]). Hereafter we always assume that $[\mathrm{SL}_{2}(\mathbb{Z})\colon G]<+\infty$ . Write $G\backslash\mathfrak{H}^{*}=G\backslash\mathfrak{H}\cup G\backslash\mathbb{P}^{1}(\mathbb{Q})$ , a disjoint union. Then $G\backslash\mathfrak{H}$ is open in $G\backslash\mathfrak{H}^{*}$ and $G\backslash\mathbb{P}^{1}(\mathbb{Q})$ is a finite set since $\mathrm{SL}_{2}(\mathbb{Z})$ acts transitively on $\mathbb{P}^{1}(\mathbb{Q})$ and hence $\lvert G\backslash\mathbb{P}^{1}(\mathbb{Q})\rvert\leq[\mathrm{SL}_{2}(\mathbb{Z})\colon G]$ . The orbits in $G\backslash\mathbb{P}^{1}(\mathbb{Q})$ are called cusps of $G\backslash\mathfrak{H}^{*}$ and by abuse of language, the points in $\mathbb{P}^{1}(\mathbb{Q})$ are also called cusps. The width of a cusp $G\cdot x\in G\backslash\mathbb{P}^{1}(\mathbb{Q})$ is defined by $w_{Gx}=w_{x}=[\overline{\mathrm{SL}_{2}(\mathbb{Z})}_{x}\colon\overline{G}_{x}]$ which is independent of the choice of the representative $x$ . We have (cf. [29, Proposition 6.3.8(b)])

[\overline{\mathrm{SL}_{2}(\mathbb{Z})}\colon\overline{G}]=\sum_{Gx\in G\backslash\mathbb{P}^{1}(\mathbb{Q})}w_{Gx}.

We also need the notion of elliptic points. It is well known that the stabilizer of each point in $\{\gamma\mathrm{i}\colon\gamma\in\mathrm{SL}_{2}(\mathbb{Z})\}$ under the action of $\mathrm{SL}_{2}(\mathbb{Z})/\{\pm I\}$ is a cyclic group of order $2$ , the stabilizer of each point in $\{\gamma\rho\colon\gamma\in\mathrm{SL}_{2}(\mathbb{Z})\}$ is a cyclic group of order $3$ where $\rho=\frac{-1+\sqrt{3}\mathrm{i}}{2}$ and the stabilizer of any other point in $\mathfrak{H}$ is trivial. Thus for $\tau\in\mathfrak{H}$ , $\lvert\overline{G}_{\tau}\rvert=1$ , $2$ or $3$ since $\lvert\overline{G}_{\tau}\rvert$ divides $\lvert\overline{\mathrm{SL}_{2}(\mathbb{Z})}_{\tau}\rvert$ . If $\lvert\overline{G}_{\tau}\rvert=2$ ( $3$ respectively) then we call $p(\tau)$ an elliptic point for $G\backslash\mathfrak{H}^{*}$ of period $2$ ( $3$ respectively). By abuse of language, we also call the number $\tau$ an elliptic point if $p(\tau)$ is.

The description and verification of the atlas on $G\backslash\mathfrak{H}^{*}$ that turns it into a compact Riemann surface are rather technical and tedious but it is basic and well known (cf. eg. [35, Section 1.8]). Let $X_{G}$ denote the resulting compact Riemann surface. We will mainly use $X_{\Gamma_{0}(N)}$ hereafter so set $X_{0}(N)=X_{\Gamma_{0}(N)}$ for simplicity. Let $g$ be the genus of $X_{G}$ . Then by applying the Riemann-Hurwitz ramification formula to the holomorphic map $X_{G}\rightarrow X_{\mathrm{SL}_{2}(\mathbb{Z})}$ , $G\cdot z\mapsto\mathrm{SL}_{2}(\mathbb{Z})\cdot z$ , noticing that the degree of this map equals $m=[\overline{\mathrm{SL}_{2}(\mathbb{Z})}\colon\overline{G}]$ and the genus of $X_{\mathrm{SL}_{2}(\mathbb{Z})}$ is $0$ , we obtain (cf. e.g. [36, Theorem 3.1.1] for the case $G$ is a congruence subgroup)

(8)

g=1+\frac{m}{12}-\frac{\varepsilon_{2}}{4}-\frac{\varepsilon_{3}}{3}-\frac{\varepsilon_{\infty}}{2},

where $\varepsilon_{2}$ and $\varepsilon_{3}$ are the numbers of elliptic points of $X_{G}$ of period $2$ and $3$ respectively and $\varepsilon_{\infty}$ is the number of cusps of $X_{G}$ . More formally,

	$\displaystyle\varepsilon_{2}$	$\displaystyle=\mathop{\mathrm{\#}}\{G\cdot z\in G\backslash\mathrm{SL}_{2}(\mathbb{Z})\mathrm{i}\colon\lvert\overline{G}_{z}\rvert=2\},$
	$\displaystyle\varepsilon_{3}$	$\displaystyle=\mathop{\mathrm{\#}}\{G\cdot z\in G\backslash\mathrm{SL}_{2}(\mathbb{Z})\rho\colon\lvert\overline{G}_{z}\rvert=3\},\quad\rho=\frac{-1+\sqrt{3}\mathrm{i}}{2},$
	$\displaystyle\varepsilon_{\infty}$	$\displaystyle=\mathop{\mathrm{\#}}G\backslash\mathbb{P}^{1}(\mathbb{Q}).$

For $G=\Gamma_{0}(N)$ , it is known that

(9)	$\displaystyle m$	$\displaystyle=[\overline{\mathrm{SL}_{2}(\mathbb{Z})}\colon\overline{\Gamma_{0}(N)}]=[\mathrm{SL}_{2}(\mathbb{Z})\colon\Gamma_{0}(N)]=N\prod_{p\mid N}\left(1+\frac{1}{p}\right),$
(10)	$\displaystyle\varepsilon_{2}$	$\displaystyle=\begin{dcases}\prod_{p\mid N}\left(1+\genfrac{(}{)}{}{}{-4}{p}\right)&\text{ if }4\nmid N,\\ 0&\text{ if }4\mid N,\end{dcases}$
(11)	$\displaystyle\varepsilon_{3}$	$\displaystyle=\begin{dcases}\prod_{p\mid N}\left(1+\genfrac{(}{)}{}{}{-3}{p}\right)&\text{ if }9\nmid N,\\ 0&\text{ if }9\mid N,\end{dcases}$
(12)	$\displaystyle\varepsilon_{\infty}$	$\displaystyle=\sum_{d\mid N}\phi(d,N/d),$

where $p$ denotes a prime, $\phi(d,N/d)=\phi(\gcd(d,N/d))$ , and $\phi$ is the Euler function. For proofs, see [29, Corollary 6.2.13, Corollary 6.3.24(b)] and [36, Corollary 3.7.2].

Now let us associate a $\mathbb{Q}$ -valued divisor to any meromorphic modular form of rational weight. For integral or half-integral weights, this association is the same as the ordinary one⁵⁵5But it is different from the one described in [33, p. 299] (cf. [36, eq. (3.2) and (3.3)]). Our treatment has the feature that one need not distinguish between regular and irregular cusps.

Definition 3.1.

Let $G$ be a subgroup of $\mathrm{SL}_{2}(\mathbb{Z})$ of finite index, $D\in\mathbb{Z}_{\geq 1}$ and $k\in\frac{1}{D}\mathbb{Z}$ . Let $\chi$ be a multiplier system for $G$ of weight $k$ of cover index $D$ (not assumed to be of finite order). Let $f$ be a nonzero meromorphic modular form of weight $k$ for $G$ with character $\chi$ . We define its divisor $\mathop{\mathrm{div}}(f)=\sum_{x\in X_{G}}\mathop{\mathrm{div}}_{x}(f)\cdot(x)\in\mathscr{D}_{\mathbb{Q}}(X_{G})$ as follows:

(a)

If $\tau\in\mathfrak{H}$ , then set $\mathop{\mathrm{div}}_{G\tau}(f)=n_{0}/\lvert\overline{G}_{\tau}\rvert$ where $n_{0}$ is the integer such that $f(z)=\sum_{n\geq n_{0}}c_{n}(z-\tau)^{n}$ , $c_{n_{0}}\neq 0$ .
(b)

If $x\in\mathbb{P}^{1}(\mathbb{Q})$ and $\gamma\in\mathrm{SL}_{2}(\mathbb{Z})$ satisfying $\gamma(\mathrm{i}\infty)=x$ , then set $\mathop{\mathrm{div}}_{Gx}(f)=\mathop{\mathrm{ord}}_{\mathrm{i}\infty}(f|_{k}\widetilde{\gamma})\cdot w_{Gx}$ .

One can verify that $\mathop{\mathrm{div}}(f)$ is well-defined, which is not totally trivial but straightforward. The usefulness of this notion relies on the following two simple facts:

•

If $g$ is another nonzero meromorphic modular form of weight $k^{\prime}\in\frac{1}{D}\mathbb{Z}$ for $G$ with character $\chi^{\prime}$ , then we have $\mathop{\mathrm{div}}(fg)=\mathop{\mathrm{div}}(f)+\mathop{\mathrm{div}}(g)$ in $\mathscr{D}_{\mathbb{Q}}(X_{G})$ (cf. Remark 2.3).
•

If $k=0$ and $\chi$ is the trivial character, then $f$ descends to a meromorphic function $\widetilde{f}$ on $X_{G}$ . We have $\mathop{\mathrm{div}}(f)=\mathop{\mathrm{div}}(\widetilde{f})$ where $\mathop{\mathrm{div}}(\widetilde{f})$ is the usual divisor of a meromorphic function.

Note that the factors $1/\lvert\overline{G}_{\tau}\rvert$ and $w_{Gx}$ in Definition 3.1 are chosen to let the latter fact hold (to see this one must dive into the atlas of $X_{G}$ which we have omitted). Also note that the former fact holds because the factors $1/\lvert\overline{G}_{\tau}\rvert$ and $w_{Gx}$ remain unchanged for all $k$ .

Similarly, we have $\mathop{\mathrm{div}}(f/g)=\mathop{\mathrm{div}}(f)-\mathop{\mathrm{div}}(g)$ and in particular $\mathop{\mathrm{div}}(1/g)=-\mathop{\mathrm{div}}(g)$ .

A fundamental theorem on modular forms of rational weight is the following one, which in cases of integral or half-integral weights is sometimes called the valence formula.

Theorem 3.2.

Let us use the notation of Definition 3.1 and set $m=[\overline{\mathrm{SL}_{2}(\mathbb{Z})}\colon\overline{G}]$ . Then we have

\deg(\mathop{\mathrm{div}}f)=\frac{1}{12}mk.

Proof.

Suppose that $\overline{\mathrm{SL}_{2}(\mathbb{Z})}=\bigcup_{1\leq j\leq m}\overline{G}\overline{\gamma_{j}}$ is a disjoint union where $\gamma_{j}\in\mathrm{SL}_{2}(\mathbb{Z})$ and $\overline{\gamma_{j}}=\gamma_{j}\cdot\{\pm I\}$ . Set $g=\prod_{1\leq j\leq m}f|_{k}\widetilde{\gamma_{j}}$ . Then $g|_{mk}\gamma=c_{\gamma}g$ for any $\gamma\in\widetilde{\mathrm{SL}_{2}(\mathbb{Z})^{D}}$ where $c_{\gamma}\in\mathbb{C}^{\times}$ . Since $g$ is not identically zero, the map $\gamma\mapsto c_{\gamma}$ is a linear character on $\widetilde{\mathrm{SL}_{2}(\mathbb{Z})^{D}}$ . According to a presentation of $\widetilde{\mathrm{SL}_{2}(\mathbb{Z})^{D}}$ (cf. [16, Lemma 5.2]) the order of this character divides $12D$ and hence $g^{12D}$ transforms like a modular form of weight $12Dmk$ for $\mathrm{SL}_{2}(\mathbb{Z})$ with trivial character. Since $f$ is meromorphic at cusps, so is $g^{12D}$ . Therefore $g^{12D}$ is a meromorphic modular form of weight $12Dmk$ for $\mathrm{SL}_{2}(\mathbb{Z})$ with trivial character. Applying the usual valence formula for even weights (cf. [29, Theorem 5.6.1]) we obtain $\deg(\mathop{\mathrm{div}}g^{12D})=Dmk$ . It remains to prove that $12D\deg(\mathop{\mathrm{div}}f)=\deg(\mathop{\mathrm{div}}g^{12D})$ , which is equivalent to $\deg(\mathop{\mathrm{div}}f)=\deg(\mathop{\mathrm{div}}g)$ . We now prove a stronger assertion, i.e.

(13)

\mathop{\mathrm{div}}\nolimits_{x}(g)=\sum_{y\in p^{-1}(x)}\mathop{\mathrm{div}}\nolimits_{y}(f)

for any $x\in X_{\mathrm{SL}_{2}(\mathbb{Z})}$ , where $p\colon X_{G}\rightarrow X_{\mathrm{SL}_{2}(\mathbb{Z})}$ is the natural projection $G\cdot\tau\mapsto\mathrm{SL}_{2}(\mathbb{Z})\cdot\tau$ . In the case that $x\in\mathrm{SL}_{2}(\mathbb{Z})\backslash\mathfrak{H}$ , we write $x=\mathrm{SL}_{2}(\mathbb{Z})\cdot\tau$ and set $w_{\tau}=[\overline{\mathrm{SL}_{2}(\mathbb{Z})}_{\tau}\colon\overline{G}_{\tau}]$ (similar to the notion of the width of a cusp in the group-theoretical aspect). Notice that $\mathop{\mathrm{ord}}_{\tau}f|_{k}\widetilde{\gamma_{j}}=\mathop{\mathrm{ord}}_{\gamma_{j}\tau}f$ where $\mathop{\mathrm{ord}}_{\tau}f$ is the integer $n_{0}$ such that $\lim_{z\to\tau}f(z)(z-\tau)^{-n_{0}}$ is nonzero. Thus we have

\mathop{\mathrm{div}}\nolimits_{x}(g)=\lvert\overline{\mathrm{SL}_{2}(\mathbb{Z})}_{\tau}\rvert^{-1}\cdot\sum_{1\leq j\leq m}\mathop{\mathrm{ord}}\nolimits_{\gamma_{j}\tau}f.

The underlying set of the multiset $\{G\cdot\gamma_{j}\tau\colon 1\leq j\leq m\}$ equals $p^{-1}(x)$ and the multiplicity of each element $G\cdot\gamma_{j}\tau$ equals $w_{\gamma_{j}\tau}$ . It follows that

\mathop{\mathrm{div}}\nolimits_{x}(g)=\lvert\overline{\mathrm{SL}_{2}(\mathbb{Z})}_{\tau}\rvert^{-1}\cdot\sum_{y=G\cdot\tau\in p^{-1}(x)}w_{\tau}\cdot\mathop{\mathrm{ord}}\nolimits_{\tau}f=\sum_{y\in p^{-1}(x)}\mathop{\mathrm{div}}\nolimits_{y}(f),

i.e., the desired assertion (13) holds. In the other case $x\in\mathrm{SL}_{2}(\mathbb{Z})\backslash\mathbb{P}^{1}(\mathbb{Q})$ , we have $x=\mathrm{SL}_{2}(\mathbb{Z})\cdot\mathrm{i}\infty$ and its width is $1$ . Therefore

\mathop{\mathrm{div}}\nolimits_{x}(g)=\sum_{1\leq j\leq m}\mathop{\mathrm{ord}}\nolimits_{\mathrm{i}\infty}(f|_{k}\widetilde{\gamma_{j}}).

The underlying set of the multiset $\{G\cdot\gamma_{j}\mathrm{i}\infty\colon 1\leq j\leq m\}$ equals $p^{-1}(x)$ and the multiplicity of each element $G\cdot\gamma_{j}\mathrm{i}\infty$ equals $w_{\gamma_{j}\mathrm{i}\infty}$ . It follows that

\mathop{\mathrm{div}}\nolimits_{x}(g)=\sum_{y=G\cdot\gamma\mathrm{i}\infty\in p^{-1}(x)}w_{\gamma\mathrm{i}\infty}\mathop{\mathrm{ord}}\nolimits_{\mathrm{i}\infty}(f|_{k}\widetilde{\gamma})=\sum_{y\in p^{-1}(x)}\mathop{\mathrm{div}}\nolimits_{y}(f),

i.e., the desired assertion (13) holds. This concludes the proof. ∎

Remark 3.3.

The above theorem holds even when $\chi$ is non-unitary and the proof remains unchanged. For instance, one can apply it to the generalized modular forms constructed in [25].

In the rest we need to know information about the divisor of an eta-quotient times a product of Eisenstein series on $\mathrm{SL}_{2}(\mathbb{Z})$ . We let

	$\displaystyle E_{4}(\tau)$	$\displaystyle=1+240\sum_{n\in\mathbb{Z}_{\geq 1}}\sigma_{3}(n)q^{n},$
	$\displaystyle E_{6}(\tau)$	$\displaystyle=1-504\sum_{n\in\mathbb{Z}_{\geq 1}}\sigma_{5}(n)q^{n},$

where $\sigma_{k}(n)=\sum_{d\mid n}d^{k}$ . It is well known that $E_{4}\in M_{4}({\mathrm{SL}_{2}(\mathbb{Z})})$ and $E_{6}\in M_{6}({\mathrm{SL}_{2}(\mathbb{Z})})$ .

Proposition 3.4.

Let $N\in\mathbb{Z}_{\geq 1}$ , $r_{n}\in\mathbb{Q}$ for any $n\mid N$ and $t_{1},t_{2}\in\mathbb{Z}$ . Let $\chi$ be the character (6) where $D\in\mathbb{Z}_{\geq 1}$ is chosen so that $Dr_{n}\in 2\mathbb{Z}$ for any $n\mid N$ . Set

f(\tau)=\prod_{n\mid N}\eta(n\tau)^{r_{n}}\cdot E_{4}(\tau)^{t_{1}}\cdot E_{6}(\tau)^{t_{2}}.

Then $f$ is a nonzero meromorphic modular form of weight $k=\frac{1}{2}\sum_{n}r_{n}+4t_{1}+6t_{2}$ for $\Gamma_{0}(N)$ with character $\chi$ and

\deg\left([\mathop{\mathrm{div}}f]\right)=\sum_{c\mid N}\phi(c,N/c)\cdot\left[\frac{N}{24(N,c^{2})}\sum_{n\mid N}\frac{(n,c)^{2}}{n}r_{n}\right]+\left(\frac{t_{1}}{3}+\frac{t_{2}}{2}\right)m-\left\{\frac{t_{2}}{2}\right\}\varepsilon_{2}-\left\{\frac{t_{1}}{3}\right\}\varepsilon_{3},

where $m,\varepsilon_{2}$ and $\varepsilon_{3}$ are given in (9), (10) and (11).

Proof.

The fact that $f$ is a nonzero meromorphic modular form follows from Remark 2.3. It remains to compute $\deg\left([\mathop{\mathrm{div}}f]\right)$ . The divisors of $E_{4}$ and $E_{6}$ on $X_{\Gamma_{0}(N)}$ are known (cf. eg. [29, Proposition 5.6.5]) as follows:

\mathop{\mathrm{div}}\nolimits_{x}(E_{u})=\begin{dcases}1/e_{\tau}&\text{ if }x=\Gamma_{0}(N)\cdot\tau\in\Gamma_{0}(N)\backslash\mathrm{SL}_{2}(\mathbb{Z})\cdot\tau_{0},\\ 0&\text{ otherwise},\end{dcases}

where $u=4$ or $6$ , $e_{\tau}=\lvert\overline{\Gamma_{0}(N)}_{\tau}\rvert$ and $\tau_{0}=\rho=\frac{-1+\sqrt{3}\mathrm{i}}{2}$ if $u=4$ ; $\tau_{0}=\mathrm{i}$ if $u=6$ . On the other hand, if we set $f_{0}(\tau)=\prod_{n\mid N}\eta(n\tau)^{r_{n}}$ , then $\mathop{\mathrm{div}}_{x}(f_{0})=0$ for $x\in\Gamma_{0}(N)\backslash\mathfrak{H}$ since $f_{0}$ has no poles or zeros on $\mathfrak{H}$ . For $x\in\Gamma_{0}(N)\backslash\mathbb{P}^{1}(\mathbb{Q})$ , we can find $\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right)\in\mathrm{SL}_{2}(\mathbb{Z})$ such that $x=\Gamma_{0}(N)\cdot\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right)\mathrm{i}\infty$ . It is known that $w_{x}=N/(N,c^{2})$ (cf. [29, Corollary 6.3.24(a)]). It follows from this and (7) that

(14)

\mathop{\mathrm{div}}\nolimits_{x}(f_{0})=\frac{N}{24(N,c^{2})}\sum_{n\mid N}\frac{(n,c)^{2}}{n}r_{n}.

Therefore

(15)

\deg\left([\mathop{\mathrm{div}}f]\right)=\sum_{x\in\Gamma_{0}(N)\backslash\mathrm{SL}_{2}(\mathbb{Z})\rho}[t_{1}\cdot\mathop{\mathrm{div}}\nolimits_{x}(E_{4})]+\sum_{x\in\Gamma_{0}(N)\backslash\mathrm{SL}_{2}(\mathbb{Z})\mathrm{i}}[t_{2}\cdot\mathop{\mathrm{div}}\nolimits_{x}(E_{6})]+\sum_{x\in\Gamma_{0}(N)\backslash\mathbb{P}^{1}(\mathbb{Q})}[\mathop{\mathrm{div}}\nolimits_{x}(f_{0})],

where the three sums in the right-hand side are denoted by $S_{1}$ , $S_{2}$ and $S_{3}$ respectively hereafter. To compute $S_{1}$ , let $\mathrm{SL}_{2}(\mathbb{Z})=\cup_{1\leq j\leq m}\Gamma_{0}(N)\gamma_{j}$ be a disjoint union⁶⁶6Since $-I\in\Gamma_{0}(N)$ , this is equivalent to $\overline{\mathrm{SL}_{2}(\mathbb{Z})}=\cup_{1\leq j\leq m}\overline{\Gamma_{0}(N)}\overline{\gamma_{j}}$ , and as well to $\widetilde{\mathrm{SL}_{2}(\mathbb{Z})}=\cup_{1\leq j\leq m}\widetilde{\Gamma_{0}(N)}\widetilde{\gamma_{j}}$ both of which are disjoint unions.. Then the underlying set of the multiset $\{\Gamma_{0}(N)\cdot\gamma_{j}\rho\colon 1\leq j\leq m\}$ equals $\Gamma_{0}(N)\backslash\mathrm{SL}_{2}(\mathbb{Z})\rho$ and the multiplicity of each element $\Gamma_{0}(N)\cdot\gamma_{j}\rho$ equals $\lvert\overline{\mathrm{SL}_{2}(\mathbb{Z})}_{\gamma_{j}\rho}\rvert/\lvert\overline{\Gamma_{0}(N)}_{\gamma_{j}\rho}\rvert=3/e_{\gamma_{j}\rho}$ . Hence we have

S_{1}=\frac{1}{3}\sum_{1\leq j\leq m\colon e_{\gamma_{j}\rho}=1}\left[\frac{t_{1}}{e_{\gamma_{j}\rho}}\right]+\sum_{1\leq j\leq m\colon e_{\gamma_{j}\rho}=3}\left[\frac{t_{1}}{e_{\gamma_{j}\rho}}\right]=\frac{mt_{1}}{3}-\left\{\frac{t_{1}}{3}\right\}\varepsilon_{3}.

Similarly, $S_{2}=\frac{mt_{2}}{2}-\left\{\frac{t_{2}}{2}\right\}\varepsilon_{2}$ . For $S_{3}$ , we need a complete set of representatives of $\Gamma_{0}(N)\backslash\mathbb{P}^{1}(\mathbb{Q})$ , i.e., the set of $a/c\in\mathbb{P}^{1}(\mathbb{Q})$ where $c\mid N$ and for each $a_{0}$ such that $0\leq a_{0}<(c,N/c)$ with $(a_{0},c,N/c)=1$ , $a$ is chosen to satisfy $a\equiv a_{0}\bmod(c,N/c)$ and $(a,c)=1$ (cf. [29, Corollary 6.3.23]). It follows from this and (14) that

S_{3}=\sum_{c\mid N}\phi(c,N/c)\cdot\left[\frac{N}{24(N,c^{2})}\sum_{n\mid N}\frac{(n,c)^{2}}{n}r_{n}\right].

Inserting the expressions for $S_{1}$ , $S_{2}$ and $S_{3}$ into (15) gives the desired formula. ∎

4. The main theorem: Dimension formulas for rational weights

We recall the Riemann-Roch theorem here for proving the main theorem. Let $X$ be a compact Riemann surface of genus $g$ and $D\in\mathscr{D}_{\mathbb{Q}}(X)$ . Then the Riemann-Roch space is defined by $\mathscr{L}(D)=\{f\in\mathscr{M}(X)\colon\mathop{\mathrm{div}}(f)\geq-D\}$ with the convention that $0\in\mathscr{L}(D)$ . It is immediate that $\mathscr{L}(D)$ is a complex vector space and that $\mathscr{L}(D)=\mathscr{L}([D])$ . Similarly we define $\mathscr{K}(D)=\{\omega\in\mathscr{K}(X)\colon\mathop{\mathrm{div}}(\omega)\geq D\}\cup\{0\}$ . Then the Riemann-Roch theorem states that

(16)

\dim_{\mathbb{C}}\mathscr{L}(D)=\dim_{\mathbb{C}}\mathscr{K}(D)+\deg(D)-g+1,

provided that $D$ is integral, i.e., $D\in\mathscr{D}(X)$ . Note that it is tacitly understood that $\dim_{\mathbb{C}}\mathscr{L}(D)<+\infty$ . For a proof based on the existence theorems for harmonic functions on Riemann surfaces, see [33, p. 249]. Set $D=0$ ; we find that $\dim_{\mathbb{C}}\mathscr{K}(0)=g$ , that is, the dimension of the space of all holomorphic differentials is $g$ . Let $\omega_{0}$ be any nonzero meromorphic differential (necessarily exists) and $K=\mathop{\mathrm{div}}(\omega_{0})$ . Then $\mathscr{L}(K-D)\to\mathscr{K}(D)$ , $f\mapsto f\cdot\omega_{0}$ is a $\mathbb{C}$ -linear isomorphism. Now set $D=K$ in (16); we find that $\deg(K)=2g-2$ . Finally, if $D\in\mathscr{D}(X)$ and $\deg(D)>2g-2$ , then $\deg(K-D)<0$ and hence $\mathscr{L}(K-D)=\mathscr{K}(D)=\{0\}$ since the degree of a nonzero meromorphic function is $0$ . It follows that

(17)

\dim_{\mathbb{C}}\mathscr{L}(D)=\deg(D)-g+1,\quad\text{if }D\in\mathscr{D}(X),\,\deg(D)>2g-2.

Our proof of the main theorem rests on the following general lemma, which is the key point for using the Riemann-Roch theorem.

Lemma 4.1.

Let $G$ be a subgroup of $\mathrm{SL}_{2}(\mathbb{Z})$ of finite index, $D\in\mathbb{Z}_{\geq 1}$ , $k\in\frac{1}{D}\mathbb{Z}$ and $v\colon\widetilde{G}^{D}\to\mathbb{C}^{\times}$ be a character. Let $p\colon\mathfrak{H}\to G\backslash\mathfrak{H}$ be the natural projection. Suppose there exists a nonzero meromorphic modular form $f_{0}$ of weight $k$ for group $G$ with character $v$ . Then the map

	$\displaystyle\mathscr{L}([\mathop{\mathrm{div}}f_{0}])$	$\displaystyle\to M_{k}(G,v)$
	$\displaystyle f$	$\displaystyle\mapsto(f\|_{G\backslash\mathfrak{H}})\circ p\cdot f_{0}$

is a $\mathbb{C}$ -linear isomorphism. Moreover, let

(18)

R=\{x\in G\backslash\mathbb{P}^{1}(\mathbb{Q})\colon\mathop{\mathrm{div}}\nolimits_{x}(f_{0})\in\mathbb{Z}\}.

Then the map

	$\displaystyle\mathscr{L}\left([\mathop{\mathrm{div}}f_{0}]-\sum\nolimits_{x\in R}(x)\right)$	$\displaystyle\to S_{k}(G,v)$
	$\displaystyle f$	$\displaystyle\mapsto(f\|_{G\backslash\mathfrak{H}})\circ p\cdot f_{0}$

is also a $\mathbb{C}$ -linear isomorphism.

Proof.

Let $f\in\mathscr{L}([\mathop{\mathrm{div}}f_{0}])$ be arbitrary. Since $p$ is holomorphic, $(f|_{G\backslash\mathfrak{H}})\circ p$ is a modular function for group $G$ with trivial character. Thus $(f|_{G\backslash\mathfrak{H}})\circ p\cdot f_{0}$ is a meromorphic modular form of weight $k$ for group $G$ with character $v$ by Remark 2.3. Now assume $f\neq 0$ and let $x\in X_{G}$ be arbitrary. Since $f\in\mathscr{L}([\mathop{\mathrm{div}}f_{0}])$ we have $\mathop{\mathrm{div}}_{x}((f|_{G\backslash\mathfrak{H}})\circ p)=\mathop{\mathrm{div}}_{x}(f)\geq-\mathop{\mathrm{div}}_{x}(f_{0})$ . It follows that

\mathop{\mathrm{div}}\nolimits_{x}((f|_{G\backslash\mathfrak{H}})\circ p\cdot f_{0})=\mathop{\mathrm{div}}\nolimits_{x}((f|_{G\backslash\mathfrak{H}})\circ p)+\mathop{\mathrm{div}}\nolimits_{x}(f_{0})\geq 0.

Hence $(f|_{G\backslash\mathfrak{H}})\circ p\cdot f_{0}\in M_{k}(G,v)$ . It is obvious that the considered map is a $\mathbb{C}$ -linear injection. To prove the surjectivity, let $0\neq g\in M_{k}(G,v)$ be arbitrary, then by Remark 2.3 $g/f_{0}$ is a modular function for group $G$ with trivial character and hence it descends to some nonzero $f\in\mathscr{M}(X_{G})$ . For any $x\in X_{G}$ we have

\mathop{\mathrm{div}}\nolimits_{x}(f)=\mathop{\mathrm{div}}\nolimits_{x}(g/f_{0})=\mathop{\mathrm{div}}\nolimits_{x}(g)-\mathop{\mathrm{div}}\nolimits_{x}(f_{0})\geq-\mathop{\mathrm{div}}\nolimits_{x}(f_{0}).

Therefore $f\in\mathscr{L}(\mathop{\mathrm{div}}f_{0})=\mathscr{L}([\mathop{\mathrm{div}}f_{0}])$ and the image of $f$ is $g$ from which the surjectivity follows.

The assertion on the second map can be proved in a similar manner and the only thing that need to explain is that $(f|_{G\backslash\mathfrak{H}})\circ p\cdot f_{0}\in S_{k}(G,v)$ if and only if $f\in\mathscr{L}\left([\mathop{\mathrm{div}}f_{0}]-\sum\nolimits_{x\in R}(x)\right)$ . If $0\neq f\in\mathscr{L}\left([\mathop{\mathrm{div}}f_{0}]-\sum\nolimits_{x\in R}(x)\right)$ and $x\in R$ , then

\mathop{\mathrm{div}}\nolimits_{x}((f|_{G\backslash\mathfrak{H}})\circ p\cdot f_{0})=\mathop{\mathrm{div}}\nolimits_{x}(f)+\mathop{\mathrm{div}}\nolimits_{x}(f_{0})\geq 1-[\mathop{\mathrm{div}}\nolimits_{x}(f_{0})]+\mathop{\mathrm{div}}\nolimits_{x}(f_{0})>0.

On the other hand, if $x\not\in R$ then $-[\mathop{\mathrm{div}}\nolimits_{x}(f_{0})]+\mathop{\mathrm{div}}\nolimits_{x}(f_{0})>0$ and hence

\mathop{\mathrm{div}}\nolimits_{x}((f|_{G\backslash\mathfrak{H}})\circ p\cdot f_{0})=\mathop{\mathrm{div}}\nolimits_{x}(f)+\mathop{\mathrm{div}}\nolimits_{x}(f_{0})\geq-[\mathop{\mathrm{div}}\nolimits_{x}(f_{0})]+\mathop{\mathrm{div}}\nolimits_{x}(f_{0})>0.

It follows that $(f|_{G\backslash\mathfrak{H}})\circ p\cdot f_{0}\in S_{k}(G,v)$ . The converse can be proved similarly. ∎

Now we state and prove the main theorem.

Theorem 4.2.

Let $N\in\mathbb{Z}_{\geq 1}$ , $r_{n}\in\mathbb{Q}$ for any $n\mid N$ and $t\in\mathbb{Z}_{\geq 0}$ . Let $D$ be a positive integer such that $Dr_{n}\in 2\mathbb{Z}$ for any $n\mid N$ . Let $\chi\colon\widetilde{\Gamma_{0}(N)^{D}}\to\mathbb{C}^{\times}$ be the character (6) and $m$ , $\varepsilon_{2}$ , $\varepsilon_{3}$ are given by (9), (10), (11) respectively. Set $k=\frac{1}{2}\sum_{n}r_{n}+2t$ and

(19)

x_{c}=\sum_{n\mid N}\frac{N}{(N,c^{2})}\cdot\frac{(n,c)^{2}}{n}r_{n}

with $c\mid N$ . Suppose either

(20)

\sum_{c\mid N}\phi(c,N/c)\cdot\left(\left[\frac{x_{c}}{24}\right]+1\right)>0,\,t\geq 1

(21)

k>2-\frac{6}{m}\varepsilon_{2}-\frac{8}{m}\varepsilon_{3}-\frac{12}{m}\sum_{c\mid N}\phi(c,N/c)\cdot\left(1-\left\{\frac{x_{c}}{24}\right\}\right),\,t=0.

Then we have

(22)

\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(N),\chi)=\frac{k-1}{12}m+\left(\frac{1}{4}-\left\{\frac{t}{2}\right\}\right)\varepsilon_{2}+\left(\frac{1}{3}-\left\{-\frac{t}{3}\right\}\right)\varepsilon_{3}\\ +\sum_{c\mid N}\phi(c,N/c)\cdot\left(\frac{1}{2}-\left\{\frac{x_{c}}{24}\right\}\right).

Proof.

Set

f_{0}(\tau)=\prod_{n\mid N}\eta(n\tau)^{r_{n}}\cdot E_{4}(\tau)^{-t}\cdot E_{6}(\tau)^{t}.

According to Proposition 3.4 $f_{0}$ is a nonzero meromorphic modular form of weight $k$ for group $\Gamma_{0}(N)$ with character $\chi$ . Hence $\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(N),\chi)=\dim_{\mathbb{C}}\mathscr{L}([\mathop{\mathrm{div}}f_{0}])$ by Lemma 4.1. Using Proposition 3.4 again we find that

(23)

\deg\left([\mathop{\mathrm{div}}f_{0}]\right)=\sum_{c\mid N}\phi(c,N/c)\cdot\left[\frac{x_{c}}{24}\right]+\frac{t}{6}m-\left\{\frac{t}{2}\right\}\varepsilon_{2}-\left\{-\frac{t}{3}\right\}\varepsilon_{3}.

We now prove either (20) or (21) implies that $\deg\left([\mathop{\mathrm{div}}f_{0}]\right)>2g-2$ where $g$ is the genus of $X_{\Gamma_{0}(N)}$ (cf. (8)). First suppose (20) holds. If $N=1$ , then $g=0$ , $r_{1}\geq 0$ and hence

\deg\left([\mathop{\mathrm{div}}f_{0}]\right)=\left[\frac{r_{1}}{24}\right]+\frac{t}{6}-\left\{\frac{t}{2}\right\}-\left\{-\frac{t}{3}\right\}>-2=2g-2.

If $N\geq 2$ , set

\gamma(t)=\frac{t}{6}m-\left\{\frac{t}{2}\right\}\varepsilon_{2}-\left\{-\frac{t}{3}\right\}\varepsilon_{3}.

We have

\gamma(t+1)-\gamma(t)\geq\frac{1}{6}m-\frac{1}{2}\varepsilon_{2}-\frac{2}{3}\varepsilon_{3}=2g-2+\varepsilon_{\infty}

for $t\geq 1$ . Since $N\geq 2$ we have $\varepsilon_{\infty}\geq 2$ and hence $\gamma(t+1)-\gamma(t)\geq 0$ . Therefore $\deg\left([\mathop{\mathrm{div}}f_{0}]\right)$ as a function of $t\in\mathbb{Z}_{\geq 1}$ is increasing which means we need only to prove

\sum_{c\mid N}\phi(c,N/c)\cdot\left[\frac{x_{c}}{24}\right]+\frac{1}{6}m-\frac{1}{2}\varepsilon_{2}-\frac{2}{3}\varepsilon_{3}>2g-2.

This is actually equivalent to (20) according to (8) and (12). Thereby we have shown that (20) implies $\deg\left([\mathop{\mathrm{div}}f_{0}]\right)>2g-2$ . Next suppose (21) holds, which is equivalent to

(24)

\frac{1}{12}mk>\frac{1}{6}m-\frac{1}{2}\varepsilon_{2}-\frac{2}{3}\varepsilon_{3}-\varepsilon_{\infty}+\sum_{c\mid N}\phi(c,N/c)\left\{\frac{x_{c}}{24}\right\}=2g-2+\sum_{c\mid N}\phi(c,N/c)\left\{\frac{x_{c}}{24}\right\}.

Applying Theorem 3.2 and (14) to $f_{0}$ (noting $t=0$ here) and using the complete set of representatives of $\Gamma_{0}(N)\backslash\mathbb{P}^{1}(\mathbb{Q})$ described in the proof of Proposition 3.4 we find that

(25)

\frac{1}{12}m\cdot\frac{1}{2}\sum_{n}r_{n}=\deg(\mathop{\mathrm{div}}f_{0})=\sum_{c\mid N}\phi(c,N/c)\cdot\frac{x_{c}}{24}.

Inserting this into (24) we obtain $\deg\left([\mathop{\mathrm{div}}f_{0}]\right)>2g-2$ as required.

Thus, in both cases we can apply (17) to $D=[\mathop{\mathrm{div}}f_{0}]$ , $X=X_{\Gamma_{0}(N)}$ . It follows that

\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(N),\chi)=\dim_{\mathbb{C}}\mathscr{L}([\mathop{\mathrm{div}}f_{0}])=\deg([\mathop{\mathrm{div}}f_{0}])-g+1.

Inserting (23), (8) and (25) into the above identity gives the desired formula. ∎

Remark 4.3.

If neither (20) nor (21) holds, then the difference between the left-hand side and the right-hand side of $\eqref{eq:dimFormula}$ equals $\dim_{\mathbb{C}}\mathscr{K}([\mathop{\mathrm{div}}f_{0}])$ according to (16). It follows that in all cases we have

\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(N),\chi)\geq\frac{k-1}{12}m+\left(\frac{1}{4}-\left\{\frac{t}{2}\right\}\right)\varepsilon_{2}+\left(\frac{1}{3}-\left\{-\frac{t}{3}\right\}\right)\varepsilon_{3}\\ +\sum_{c\mid N}\phi(c,N/c)\cdot\left(\frac{1}{2}-\left\{\frac{x_{c}}{24}\right\}\right),

where the right-hand side is always an integer. We can derive an upper bound⁷⁷7This upper bound is still valid if $\Gamma_{0}(N)$ is replaced by any subgroup $G$ of finite index in $\mathrm{SL}_{2}(\mathbb{Z})$ . for $\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(N),\chi)$ as well. Suppose that $k\geq 0$ . (When $k<0$ $M_{k}(\Gamma_{0}(N),\chi)=\{0\}$ by Theorem 3.2.) Then

\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(N),\chi)\leq\left[\frac{1}{12}mk\right]+1,

for if $f_{1},\dots,f_{j}\in M_{k}(\Gamma_{0}(N),\chi)$ with $j>\left[\frac{1}{12}mk\right]+1$ , then by linear algebra we can find a nontrivial linear combination $g=\sum_{n}c_{n}f_{n}$ such that $\mathop{\mathrm{div}}_{x}(g)\geq\left[\frac{1}{12}mk\right]+1$ where $x\in\Gamma_{0}(N)\backslash\mathfrak{H}$ is not an elliptic point. If $g\neq 0$ then applying Theorem 3.2 to $g$ we reach a contradiction. Hence $g=0$ which means $f_{1},\dots,f_{j}$ are linearly dependent.

Remark 4.4.

We can derive a formula for $\dim_{\mathbb{C}}S_{k}(\Gamma_{0}(N),\chi)$ as well, using the second map in Lemma 4.1. However, it may happen that for certain $r_{n}$ and $t$ (22) holds while the corresponding formula for $\dim_{\mathbb{C}}S_{k}(\Gamma_{0}(N),\chi)$ is not applicable. This happens precisely when $\deg\left([\mathop{\mathrm{div}}f_{0}]\right)>2g-2$ but $\deg\left([\mathop{\mathrm{div}}f_{0}]\right)-\lvert R\rvert\leq 2g-2$ . Besides, when $k>2$ , we always have $\dim_{\mathbb{C}}S_{k}(\Gamma_{0}(N),\chi)=\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(N),\chi)-\lvert R\rvert$ which is a well known fact when $k$ is integral and $\chi$ is induced by a Dirichlet character. In the rational weight and arbitrary multiplier system case, this holds as well since the Petersson inner product can also be defined. The details are omitted here.

5. Order-character relations

Let us assume (21) holds and consider Theorem 4.2. The sequence $(x_{c})_{c\mid N}$ occurs in the formula for $\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(N),\chi)$ and is determined by the sequence $(r_{n})_{n\mid N}$ via (19). Note that $(r_{n})_{n\mid N}$ represents an eta-quotient (or its character) and $(x_{c})_{c\mid N}$ represents the orders of this eta-quotient at cusps (or its divisor). Therefore (19) can be regarded as a map that sends the character represented by $(r_{n})_{n\mid N}$ to the orders at cusps $(x_{c})_{c\mid N}$ . In some potential applications, the sequence of orders $(x_{c})_{c\mid N}$ is first given, e.g., when one want to known $\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(N),\chi)$ for fixed $N$ and $k$ , and then one works out $(r_{n})_{n\mid N}$ via (19). Motivated by this, we need an inverse formula of (19). This formula is an enhancement of a statement in [37, p. 129] and is equivalent to [21, eq. (5.13)].

Let us fix some notations in linear algebra. For $N\in\mathbb{Z}_{\geq 1}$ , $\mathscr{D}(N)$ denotes the set of positive divisors of $N$ and $\sigma_{0}(N)=\lvert\mathscr{D}(N)\rvert$ . Let $V$ be a $\mathbb{Q}$ -vector space of dimension $\sigma_{0}(N)$ , then a $\mathscr{D}(N)$ -indexed basis of $V$ is a sequence $(\mathbf{b}_{n})_{n\in\mathscr{D}(N)}$ so that $\{\mathbf{b}_{n}\}$ is a basis of $V$ . Let $W$ be a $\mathbb{Q}$ -vector space of dimension $\sigma_{0}(M)$ with a $\mathscr{D}(M)$ -indexed basis $(\mathbf{b}^{\prime}_{m})_{m\mid M}$ where $M$ is another positive integer and let $f\colon V\to W$ be a linear map. The matrix of $f$ with respect to the pair $((\mathbf{b}_{n}),(\mathbf{b}^{\prime}_{m}))$ is the $\mathscr{D}(M)\times\mathscr{D}(N)$ -indexed sequence $(a_{m,n})_{m\mid M,n\mid N}$ with the property $f(\mathbf{b}_{n})=\sum_{m\mid M}a_{m,n}\mathbf{b}^{\prime}_{m}$ . All linear algebra machinery works for such kind of matrices. The difference is just the underlying index sets: we use $\mathscr{D}(N)$ and the ordinary one is $\{1,2,\dots,\sigma_{0}(N)\}$ . We call the matrix $(a_{m,n})_{m\mid M,n\mid N}$ a $\mathscr{D}(M)\times\mathscr{D}(N)$ -indexed matrix and when $M=N$ a $\mathscr{D}(N)$ -indexed matrix.

Let $N_{1},N_{2}\in\mathbb{Z}_{\geq 1}$ such that $\gcd(N_{1},N_{2})=1$ and let $(a_{c_{1},n_{1}})_{c_{1},n_{1}\mid N_{1}}$ , $(b_{c_{2},n_{2}})_{c_{2},n_{2}\mid N_{2}}$ be $\mathscr{D}(N_{1})$ -indexed and $\mathscr{D}(N_{2})$ -indexed matrices respectively. Define

(a_{c_{1},n_{1}})_{c_{1},n_{1}\mid N_{1}}\otimes(b_{c_{2},n_{2}})_{c_{2},n_{2}\mid N_{2}}=(a_{c_{1},n_{1}}\cdot b_{c_{2},n_{2}})_{c_{1}c_{2},n_{1}n_{2}\mid N_{1}N_{2}}.

It is immediate that this is a well defined $\mathbb{Q}$ -bilinear map from $\mathbb{Q}^{\mathscr{D}(N_{1})\times\mathscr{D}(N_{1})}\times\mathbb{Q}^{\mathscr{D}(N_{2})\times\mathscr{D}(N_{2})}$ to $\mathbb{Q}^{\mathscr{D}(N_{1}N_{2})\times\mathscr{D}(N_{1}N_{2})}$ . Moreover it is universal among all bilinear maps on $\mathbb{Q}^{\mathscr{D}(N_{1})\times\mathscr{D}(N_{1})}\times\mathbb{Q}^{\mathscr{D}(N_{2})\times\mathscr{D}(N_{2})}$ and hence is a tensor product. We call it the Kronecker product.

Following Bhattacharya [21] we define $A_{N}=\left(\frac{N}{(N,c^{2})}\cdot\frac{(n,c)^{2}}{n}\right)_{c,n\mid N}$ so that (19) becomes $x_{c}=\sum_{n\mid N}A_{N}(c,n)r_{n}$ . It is immediate that $A_{N_{1}N_{2}}=A_{N_{1}}\otimes A_{N_{2}}$ whenever $\gcd(N_{1},N_{2})=1$ .

Proposition 5.1.

Let $N\in\mathbb{Z}_{\geq 1}$ , $(x_{c})_{c\mid N}$ and $(r_{n})_{n\mid N}$ be in $\mathbb{Q}^{\mathscr{D}(N)}$ . Then the following two relations are equivalent:

(26)		$\displaystyle x_{c}$	$\displaystyle=\sum_{n\mid N}\frac{N}{(N,c^{2})}\cdot\frac{(n,c)^{2}}{n}r_{n},\quad\forall c\mid N,$
(27)		$\displaystyle r_{n}$	$\displaystyle=\frac{1}{N}\prod_{p\mid N}\frac{p}{p^{2}-1}\cdot\sum_{c\mid N}\left(\prod_{p^{\alpha}\parallel N}b_{p,\alpha}(v_{p}(n),v_{p}(c))\right)x_{c},\quad\forall n\mid N,$

where $p$ denotes a prime, $v_{p}(n)$ is the integer $\beta$ such that $p^{\beta}\parallel n$ and

b_{p,\alpha}(i,j)=\begin{dcases}p&\text{if }i=j=0\text{ or }\alpha,\\ (p^{2}+1)p^{\min(i,\alpha-i)-1}&\text{if }0<i=j<\alpha,\\ -p^{\min(j,\alpha-j)}&\text{if }\lvert i-j\rvert=1,\\ 0&\text{otherwise}.\end{dcases}

In another words, we have

A_{N}^{-1}(n,c)=\frac{1}{N}\prod_{p\mid N}\frac{p}{p^{2}-1}\cdot\prod_{p^{\alpha}\parallel N}b_{p,\alpha}(v_{p}(n),v_{p}(c)).

This proposition is due to Bhattacharya (cf. [21, eq. (5.13)]).

Proof.

Suppose $N_{1},N_{2}\in\mathbb{Z}_{\geq 1}$ such that $\gcd(N_{1},N_{2})=1$ . Then $A_{N_{1}}$ and $A_{N_{2}}$ are both invertible if and only if $A_{N_{1}}\otimes A_{N_{2}}$ is and in this case $(A_{N_{1}}\otimes A_{N_{2}})^{-1}=A_{N_{1}}^{-1}\otimes A_{N_{2}}^{-1}$ by basic properties of the Kronecker product. Therefore it is sufficient to prove that

A_{p^{\alpha}}^{-1}(p^{i},p^{j})=\frac{1}{p^{\alpha-1}(p^{2}-1)}\cdot b_{p,\alpha}(i,j),

where $p$ is a prime, $\alpha\in\mathbb{Z}_{\geq 1}$ and $0\leq i,j\leq\alpha$ . This is equivalent to

(28)		$\displaystyle\sum_{j=0}^{\alpha}A_{p^{\alpha}}(p^{i},p^{j})\cdot b_{p,\alpha}(j,i)=(p^{2}-1)p^{\alpha-1},$
(29)		$\displaystyle\sum_{j=0}^{\alpha}A_{p^{\alpha}}(p^{i_{1}},p^{j})\cdot b_{p,\alpha}(j,i_{2})=0\text{ if }i_{1}\neq i_{2}.$

Note that

A_{p^{\alpha}}(p^{i},p^{j})=\frac{p^{\alpha}}{(p^{\alpha},p^{2i})}\cdot\frac{(p^{j},p^{i})^{2}}{p^{j}}=\begin{dcases}p^{\alpha-j}&\text{ if }i\leq j\text{ and }i\leq\alpha/2,\\ p^{2i-j}&\text{ if }i\leq j\text{ and }i>\alpha/2,\\ p^{\alpha-2i+j}&\text{ if }i\geq j\text{ and }i\leq\alpha/2,\\ p^{j}&\text{ if }i\geq j\text{ and }i>\alpha/2.\end{dcases}

Now we begin to prove (28). In the case $i=0$ , we have

	$\displaystyle\sum_{j=0}^{\alpha}A_{p^{\alpha}}(p^{i},p^{j})\cdot b_{p,\alpha}(j,i)$	$\displaystyle=A_{p^{\alpha}}(p^{0},p^{0})\cdot b_{p,\alpha}(0,0)+A_{p^{\alpha}}(p^{0},p^{1})\cdot b_{p,\alpha}(1,0)$
		$\displaystyle=p^{\alpha}\cdot p+p^{\alpha-1}\cdot(-1)=(p^{2}-1)p^{\alpha-1}$

as required. In the case $i=\alpha$ , we have

	$\displaystyle\sum_{j=0}^{\alpha}A_{p^{\alpha}}(p^{i},p^{j})\cdot b_{p,\alpha}(j,i)$	$\displaystyle=A_{p^{\alpha}}(p^{\alpha},p^{\alpha-1})\cdot b_{p,\alpha}(\alpha-1,\alpha)+A_{p^{\alpha}}(p^{\alpha},p^{\alpha})\cdot b_{p,\alpha}(\alpha,\alpha)$
		$\displaystyle=p^{\alpha-1}\cdot(-1)+p^{\alpha}\cdot p=(p^{2}-1)p^{\alpha-1}$

as required. In the case $0<i\leq\alpha/2$ we have

		$\displaystyle\sum_{j=0}^{\alpha}A_{p^{\alpha}}(p^{i},p^{j})\cdot b_{p,\alpha}(j,i)$
	$\displaystyle=$	$\displaystyle A_{p^{\alpha}}(p^{i},p^{i-1})\cdot b_{p,\alpha}(i-1,i)+A_{p^{\alpha}}(p^{i},p^{i})\cdot b_{p,\alpha}(i,i)+A_{p^{\alpha}}(p^{i},p^{i+1})\cdot b_{p,\alpha}(i+1,i)$
	$\displaystyle=$	$\displaystyle p^{\alpha-i-1}\cdot(-p^{\min(i,\alpha-i)})+p^{\alpha-i}\cdot(p^{2}+1)p^{\min(i,\alpha-i)-1}+p^{\alpha-i-1}\cdot(-p^{\min(i,\alpha-i)})$
	$\displaystyle=$	$\displaystyle(p^{2}-1)p^{\alpha-1}$

as required. In the remaining case $\alpha/2<i<\alpha$ we have

		$\displaystyle\sum_{j=0}^{\alpha}A_{p^{\alpha}}(p^{i},p^{j})\cdot b_{p,\alpha}(j,i)$
	$\displaystyle=$	$\displaystyle A_{p^{\alpha}}(p^{i},p^{i-1})\cdot b_{p,\alpha}(i-1,i)+A_{p^{\alpha}}(p^{i},p^{i})\cdot b_{p,\alpha}(i,i)+A_{p^{\alpha}}(p^{i},p^{i+1})\cdot b_{p,\alpha}(i+1,i)$
	$\displaystyle=$	$\displaystyle p^{i-1}\cdot(-p^{\min(i,\alpha-i)})+p^{i}\cdot(p^{2}+1)p^{\min(i,\alpha-i)-1}+p^{i-1}\cdot(-p^{\min(i,\alpha-i)})$
	$\displaystyle=$	$\displaystyle(p^{2}-1)p^{\alpha-1}$

which concludes the proof of (28). We can prove (29) in a similar manner which is tedious (there are so many cases) so we omit the details. ∎

Remark 5.2.

There is another useful formula concerning the matrix $A_{N}$ :

\sum_{c\mid N}\phi(c,N/c)\frac{N}{(N,c^{2})}\frac{(n,c)^{2}}{n}=N\prod_{p\mid N}\left(1+\frac{1}{p}\right).

In another words, the Euclidean inner product of each column of $A_{N}$ and $(\phi(c,N/c))_{c\mid N}$ is equal to $N\prod_{p\mid N}\left(1+\frac{1}{p}\right)$ . This can be proved by applying Theorem 3.2 to $\eta(n\tau)$ . As a consequence, $\sum_{c\mid N}\phi(c,N/c)x_{c}=N\prod_{p\mid N}\left(1+\frac{1}{p}\right)\cdot\sum_{n\mid N}r_{n}$ .

Example 5.3.

We present some examples concerning dimensions of spaces of weight $2$ for infinite many unitary characters. Suppose that $N$ is square-free for simplicity. Let $\mathbf{x}=(x_{c})_{c\mid N}\in\mathbb{Q}^{\mathscr{D}(N)}$ such that $\sum_{c\mid N}x_{c}=0$ . Then (20) with $t=1$ holds since $\phi(c,N/c)=1$ in the current case. Thus (22) holds. According to the above remark we have $\sum_{n\mid N}r_{n}=m^{-1}\sum_{c\mid N}x_{c}=0$ and hence $k=2$ . Therefore (22) is equivalent to

(30)

\dim_{\mathbb{C}}M_{2}(\Gamma_{0}(N),\chi)=g-1+\sigma_{0}(N)+\sum_{c\mid N}\left[\frac{x_{c}}{24}\right],

where $g$ is the genus of $X_{\Gamma_{0}(N)}$ . The multiplier system $\chi$ is the character (6) where $(r_{n})_{n\mid N}$ is determined by $\mathbf{x}$ via (27) which in the current case is

(31)

r_{n}=\prod_{p\mid N}(p^{2}-1)^{-1}\cdot\sum_{c\mid N}\left((-1)^{\mathop{\mathrm{\#}}\{p\mid N\colon v_{p}(n)\neq v_{p}(c)\}}\cdot\prod_{p\mid N,\,v_{p}(n)=v_{p}(c)}p\right)x_{c}.

More generally, if $N\in\mathbb{Z}_{\geq 1}$ is not divisible by $16$ and has no odd square factor except $1$ , then we have $\phi(c,N/c)=1$ for any $c\mid N$ as well and hence (30) is still valid. However in this case, (31) should be modified slightly at the prime factor $2$ .

Now we try to find out one-dimensional spaces among the above spaces. Note that

g-1<\dim_{\mathbb{C}}M_{2}(\Gamma_{0}(N),\chi)\leq g-1+\sigma_{0}(N),

so it is necessary $g=0$ or $1$ and $N>1$ . When $g=0$ , since $N\in\mathbb{Z}_{\geq 2}$ is not divisible by $16$ and has no odd square factor except $1$ , then $N\in\{2,3,4,5,6,7,8,10,12,13\}$ . For such $N$ $\dim_{\mathbb{C}}M_{2}(\Gamma_{0}(N),\chi)=1$ if $\mathbf{x}$ satisfies

\sum_{c\mid N}x_{c}=0,\quad\sum_{c\mid N}\left[\frac{x_{c}}{24}\right]=2-\sigma_{0}(N).

When $g=1$ , we have $N\in\{11,14,15,17,19,20,21,24\}$ . For such $N$ , $\dim_{\mathbb{C}}M_{2}(\Gamma_{0}(N),\chi)=1$ if $\mathbf{x}$ satisfies

\sum_{c\mid N}x_{c}=0,\quad\sum_{c\mid N}\left[\frac{x_{c}}{24}\right]=1-\sigma_{0}(N).

Let us look at a very simple case $N=11$ . Since $\sigma_{0}(11)=2$ , $\dim_{\mathbb{C}}M_{2}(\Gamma_{0}(11),\chi)=1$ if and only if $x_{1}\not\in 24\mathbb{Z}$ (of course $x_{11}=-x_{1}$ ). We have $r_{1}=-r_{11}=x_{1}/10$ so $\chi$ is the multiplier system of $\left(\eta(\tau)/\eta(11\tau)\right)^{x_{1}/10}$ which descends to a unitary character on $\Gamma_{0}(11)$ . If $x_{1}\in 24\mathbb{Z}$ , then $\dim_{\mathbb{C}}M_{2}(\Gamma_{0}(11),\chi)=2$ . There are two completely different subcases in this case: $x_{1}\in 120\mathbb{Z}$ and $x_{1}\not\in 120\mathbb{Z}$ . If $x_{1}\in 120\mathbb{Z}$ , then $\chi$ is trivial and the formula reads $\dim_{\mathbb{C}}M_{2}(\Gamma_{0}(11))=2$ which is well known (cf. [29, Corollary 7.4.3]). Otherwise, if $x_{1}\not\in 120\mathbb{Z}$ then $x_{1}\not\in 10\mathbb{Z}$ , which implies that $\chi$ is the character of an eta-quotient of rational exponents. To our best knowledge, the dimension formulas in such situation are new. Moreover, it seems that $\ker\chi$ is a noncongruence subgroup of $\mathrm{SL}_{2}(\mathbb{Z})$ so that our formulas may be applied to the theory of noncongruence modular forms.

6. The classification of characters induced by eta-quotients

In this section we give a complete classification of the characters of eta-quotients of level $N$ and cover index $2$ for any $N\in\mathbb{Z}_{\geq 1}$ . The motivation is that, with this classification in hand, we can know exactly the set of $\chi$ for which $\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(N),\chi)$ can be calculated using Theorem 4.2. Since we only deal with the cover index $D=2$ , set $\widetilde{\Gamma_{0}(N)}=\widetilde{\Gamma_{0}(N)^{2}}$ in this section.

Now we establish some notations. Each $(r_{n})_{n\mid N}\in\mathbb{Z}^{\mathscr{D}(N)}$ represents an eta-quotient $\prod_{n\mid N}\eta(n\tau)^{r_{n}}$ of level $N$ and cover index $2$ bijectively and $(r_{n})_{n\mid N}$ with $\sum_{n}r_{n}=2k$ represents an eta-quotient of weight $k$ . Thus, set

S(N,k)=\left\{(r_{n})_{n\mid N}\in\mathbb{Z}^{\mathscr{D}(N)}\colon\sum\nolimits_{n\mid N}r_{n}=2k\right\},\quad N\in\mathbb{Z}_{\geq 1},\,k\in\frac{1}{2}\mathbb{Z},

and $S(N)=S(N,0)$ . Then $\mathbb{Z}^{\mathscr{D}(N)}=\cup_{2k\in\mathbb{Z}}S(N,k)$ which is a disjoint union. In fact, $S(N)$ is a submodule of $\mathbb{Z}^{\mathscr{D}(N)}$ and each $S(N,k)$ is a coset in $\mathbb{Z}^{\mathscr{D}(N)}/S(N)$ . The element $(r_{n})_{n\mid N}$ in $\mathbb{Z}^{\mathscr{D}(N)}$ is customary to be written as $\mathbf{r}=\sum_{n\mid N}r_{n}e_{n}$ where $\{e_{n}\colon n\mid N\}$ is the standard basis of $\mathbb{Z}^{\mathscr{D}(N)}$ . Moreover, the character (6) with $D=2$ is denoted by $\chi_{\mathbf{r}}$ to emphasize its dependence upon $\mathbf{r}\in S(N,k)$ .

Two different vectors $\mathbf{r}$ , $\mathbf{r}^{\prime}$ in $S(N,k)$ may correspond to the same character $\chi_{\mathbf{r}}=\chi_{\mathbf{r}^{\prime}}$ . The following lemma of Newman [38] tells us precisely when this happens.

Lemma 6.1.

Let $N\in\mathbb{Z}_{\geq 1}$ , $k\in\frac{1}{2}\mathbb{Z}$ . Let $\mathbf{r}=(r_{n})_{n\mid N}$ and $\mathbf{r}^{\prime}=(r^{\prime}_{n})_{n\mid N}$ be vectors in $S(N,k)$ . Then $\chi_{\mathbf{r}}=\chi_{\mathbf{r}^{\prime}}$ if and only if

	$\displaystyle\sum_{n\mid N}n\cdot(r_{n}-r^{\prime}_{n})\equiv 0\pmod{24},$
	$\displaystyle\sum_{n\mid N}\frac{N}{n}\cdot(r_{n}-r^{\prime}_{n})\equiv 0\pmod{24},$
	$\displaystyle\prod_{2\nmid r_{n}-r^{\prime}_{n}}n\text{ is a perfect square}.$

Proof.

This is a special case of the last assertion of [20, Theorem 3.9]. Setting $l=1$ in that assertion and noting that $\chi_{\mathbf{r}}=\chi_{\mathbf{r}^{\prime}}$ if and only if they are $\widetilde{I}$ -compatible give the desired equivalence. ∎

Remark 6.2.

It seems that this method does not work in the case $D\geq 3$ since we can not find any generalized double coset operator (cf. [20, Definition 1]) between characters of eta-quotients of cover index $D\geq 3$ .

As a corollary, for $\mathbf{r}=(r_{n})_{n\mid N}\in S(N)$ , $\chi_{\mathbf{r}}$ is trivial if and only if $\sum_{n}n\cdot r_{n}\equiv 0\bmod{24}$ , $\sum_{n}\frac{N}{n}\cdot r_{n}\equiv 0\bmod{24}$ and $\prod_{2\nmid r_{n}}n$ is a perfect square. The set of all such $\mathbf{r}$ is denoted by $Z(N)$ which is a subgroup of $S(N)$ . We define

\mathscr{S}(N,k)=S(N,k)/Z(N)=\left\{\mathbf{r}+Z(N)\colon\mathbf{r}\in S(N,k)\right\},

and $\mathscr{S}(N)=\mathscr{S}(N,0)$ . Thus $\mathbb{Z}^{\mathscr{D}(N)}/Z(N)=\cup_{2k\in\mathbb{Z}}\mathscr{S}(N,k)$ , which is a disjoint union. Note that elements of $\mathscr{S}(N,k)$ are in one-to-one correspondence with characters of eta-quotients of level $N$ , cover index $2$ and weight $k$ according to Lemma 6.1, so to classify such characters we need only to obtain a complete system of representatives of $\mathscr{S}(N,k)$ . Moreover, the map $\mathscr{S}(N)\to\mathscr{S}(N,k)$ that sends $\sum_{n\mid N}r_{n}e_{n}+Z(N)$ to $2ke_{1}+\sum_{n\mid N}r_{n}e_{n}+Z(N)$ is a bijection. Therefore, we have reduced our task to the study of $\mathscr{S}(N)$ . (For the result, see Theorem 6.8.)

Remark 6.3.

Let $\chi_{\mathbf{r}}\colon\widetilde{\Gamma_{0}(N)}\to\mathbb{C}^{\times}$ be a character given by (6) where $\mathbf{r}\in S(N)$ . Then $\chi_{\mathbf{r}}$ is the multiplier system of an eta-quotient of weight $0$ and cover index $2$ . According to the third property in Remark 2.2, $\chi_{\mathbf{r}}$ descends to a character on $\Gamma_{0}(N)$ . Thus $\mathscr{S}(N)$ can be regarded as a subgroup of the group of unitary linear characters of $\Gamma_{0}(N)$ .

The sets given in the following definition are subsets of $\mathscr{D}(N)$ that occur in the complete system of representatives of $\mathscr{S}(N)$ .

Definition 6.4.

Let $N$ be a positive integer and $p$ be a prime divisor of $N$ . If $p\neq 2,3$ then define $\mathscr{B}_{N}^{p}=\{p\}$ . For $p=2$ , we define

\mathscr{B}_{N}^{2}=\begin{dcases}\{2^{1}\}&\text{ if }v_{2}(N)=1,\\ \{2^{1},2^{2}\}&\text{ if }v_{2}(N)=2,\\ \{2^{\alpha-2},2^{\alpha-1},2^{\alpha}\}&\text{ if }v_{2}(N)=\alpha\geq 3.\\ \end{dcases}

For $p=3$ , we define

\mathscr{B}_{N}^{3}=\begin{dcases}\{3^{1}\}&\text{ if }v_{3}(N)=1,\\ \{3^{\alpha-1},3^{\alpha}\}&\text{ if }v_{3}(N)=\alpha\geq 2.\\ \end{dcases}

Finally, set $\mathscr{B}_{N}=\cup_{p\mid N}\mathscr{B}_{N}^{p}$ . Note that $\mathscr{B}_{1}=\emptyset$ .

We introduce a useful notation which will be used in the proofs below: if $\mathbf{r}_{1},\mathbf{r}_{2}\in S(N)$ , then $\mathbf{r}_{1}\sim_{N}\mathbf{r}_{2}$ , or simply $\mathbf{r}_{1}\sim\mathbf{r}_{2}$ , means $\mathbf{r}_{1}-\mathbf{r}_{2}\in Z(N)$ . This is an equivalence relation compatible with the addition of $S(N)$ .

Lemma 6.5.

Let $N=2^{\alpha_{0}}\cdot 3^{\alpha_{1}}\cdot N_{1}$ with $\gcd(N_{1},6)=1$ and $\alpha_{0},\alpha_{1}\in\mathbb{Z}_{\geq 0}$ .

•

For any $0\leq\beta_{0}\leq\alpha_{0}$ , $0\leq\beta_{1}\leq\alpha_{1}$ and $n_{1},n_{2}\mid N_{1}$ with $\gcd(n_{1},n_{2})=1$ , there exist $x,y,z\equiv 1\bmod 2$ such that

(32)

e_{2^{\beta_{0}}3^{\beta_{1}}n_{1}n_{2}}\sim xe_{2^{\beta_{0}}3^{\beta_{1}}n_{1}}+ye_{2^{\beta_{0}}3^{\beta_{1}}n_{2}}+ze_{2^{\beta_{0}}3^{\beta_{1}}}.

•

For any $0\leq\beta_{0}\leq\alpha_{0}$ , $0\leq\beta_{1}\leq\alpha_{1}$ and $n\mid N_{1}$ , there exist $x,y\equiv 1\bmod 2$ and $z\in\mathbb{Z}$ such that

(33) $e_{2^{\beta_{0}}3^{\beta_{1}}n}\sim xe_{2^{\beta_{0}}3^{\beta_{1}}}+ye_{n}+ze_{1}.$
•

For any $0\leq\beta_{0}\leq\alpha_{0}$ , $0\leq\beta_{1}\leq\alpha_{1}$ , there exist $x,y,z\in\mathbb{Z}$ such that

(34) $e_{2^{\beta_{0}}3^{\beta_{1}}}\sim xe_{2^{\beta_{0}}}+ye_{3^{\beta_{1}}}+ze_{1}.$

Proof.

First we prove (32). If $n_{1}=1$ or $n_{2}=1$ then (32) is immediate. Thus assume that $n_{1},n_{2}>1$ . By the definition of $Z(N)$ , a solution $(x,y,z)\in\mathbb{Z}^{3}$ of the system

(35)

\begin{dcases}2^{\beta_{0}}3^{\beta_{1}}n_{1}x+2^{\beta_{0}}3^{\beta_{1}}n_{2}y+2^{\beta_{0}}3^{\beta_{1}}z\equiv 2^{\beta_{0}}3^{\beta_{1}}n_{1}n_{2}\pmod{24}\\ \frac{N}{2^{\beta_{0}}3^{\beta_{1}}n_{1}}x+\frac{N}{2^{\beta_{0}}3^{\beta_{1}}n_{2}}y+\frac{N}{2^{\beta_{0}}3^{\beta_{1}}}z\equiv\frac{N}{2^{\beta_{0}}3^{\beta_{1}}n_{1}n_{2}}\pmod{24}\\ x,y,z\equiv 1\pmod{2}\end{dcases}

is also a solution of (32). Moreover, a solution of the system

(36)

\begin{dcases}n_{1}x+n_{2}y+z\equiv n_{1}n_{2}\pmod{24}\\ n_{2}x+n_{1}y+n_{1}n_{2}z\equiv 1\pmod{24}\\ x,y,z\equiv 1\pmod{2}\end{dcases}

is a solution of (35). Since $2,3\nmid n_{1},n_{2}$ we have $n_{1}^{2}\equiv 1\bmod 24$ and $n_{2}^{2}\equiv 1\bmod 24$ . Thus the first equation of (36) is equivalent to the second one. Therefore let $x,y\equiv 1\bmod 2$ be arbitrary and set $z\equiv n_{1}n_{2}-n_{1}x-n_{2}y$ ; we obtain a solution of (36) and hence of (32).

The proofs of (33) and (34) are similar of which we omit the details. ∎

Lemma 6.6.

Any element in $\mathscr{S}(N)$ can be represented as

(37)

\sum_{n\in\mathscr{B}_{N}}c_{n}e_{n}-\left(\sum_{n\in\mathscr{B}_{N}}c_{n}\right)e_{1}+Z(N),

where $c_{n}\in\mathbb{Z}$ and $0\leq c_{n}<24$ for any $n\in\mathscr{B}_{N}$ .

As a consequence, $\mathscr{S}(N)$ is a finite group and $\lvert\mathscr{S}(N)\rvert\leq 24^{\lvert\mathscr{B}_{N}\rvert}$ .

Proof.

Let $\mathbf{r}+Z(N)=\sum r_{n}e_{n}+Z(N)\in\mathscr{S}(N)$ be arbitrary. According to Lemma 6.5 we can assume that $r_{n}=0$ unless $n$ is a prime power or $n=1$ . Therefore there exist $\mathbf{r}^{(p)}+Z(N)=\sum_{0\leq\beta\leq v_{p}(N)}r^{(p)}_{p^{\beta}}e_{p^{\beta}}+Z(N)\in\mathscr{S}(N)$ such that $\mathbf{r}\sim\sum_{p\mid N}\mathbf{r}^{(p)}$ . (Explicitly, let $r^{(p)}_{p^{\beta}}=r_{p^{\beta}}$ for $\beta>0$ and $r^{(p)}_{p^{0}}=-\sum_{\beta>0}r_{p^{\beta}}$ .) It remains to find $c_{n}\in\mathbb{Z}$ such that

(38)

\mathbf{r}^{(p)}\sim\sum_{n\in\mathscr{B}_{N}^{p}}c_{n}e_{n}-\left(\sum_{n\in\mathscr{B}_{N}^{p}}c_{n}\right)e_{1}

and $0\leq c_{n}<24$ for each $p\mid N$ . If $p\neq 2,3$ then (38) is equivalent to

\begin{dcases}\sum_{1\leq\beta\leq v_{p}(N)}(p^{\beta}-1)r^{(p)}_{p^{\beta}}\equiv(p-1)c_{p}\pmod{24}\\ \sum_{1\leq\beta\leq v_{p}(N)}(Np^{-\beta}-N)r^{(p)}_{p^{\beta}}\equiv(Np^{-1}-N)c_{p}\pmod{24}\\ c_{p}\equiv\sum_{\beta\geq 1,\,r^{(p)}_{p^{\beta}}\equiv 1\bmod 2}\beta\pmod{2}.\end{dcases}

Since $p^{2}\equiv 1\bmod{24}$ , the solution of this system is $c_{p}\equiv\sum_{2\nmid\beta}r^{(p)}_{p^{\beta}}\bmod{24/(12,p-1)}$ . If $p=2$ and $v_{2}(N)=\alpha\geq 3$ , then (38) is equivalent to

\begin{dcases}\sum_{1\leq\beta\leq\alpha}(2^{\beta}-1)r^{(2)}_{2^{\beta}}\equiv(2^{\alpha}-1)c_{2^{\alpha}}+(2^{\alpha-1}-1)c_{2^{\alpha-1}}+(2^{\alpha-2}-1)c_{2^{\alpha-2}}\pmod{24}\\ \sum_{1\leq\beta\leq\alpha}(N2^{-\beta}-N)r^{(2)}_{2^{\beta}}\equiv(N2^{-\alpha}-N)c_{2^{\alpha}}+(N2^{-\alpha+1}-N)c_{2^{\alpha-1}}+(N2^{-\alpha+2}-N)c_{2^{\alpha-2}}\pmod{24}\\ \sum_{\begin{subarray}{c}{\beta\in\{\alpha-2,\alpha-1,\alpha\}}\\ {c_{2^{\beta}}\equiv r^{(2)}_{2^{\beta}}+1\bmod 2}\end{subarray}}\beta\equiv\sum_{1\leq\beta<\alpha-2,\,r^{(2)}_{2^{\beta}}\equiv 1\bmod 2}\beta\pmod{2}.\end{dcases}

By considering the subcases $2\mid\alpha\geq 6$ , $2\nmid\alpha\geq 5$ and $\alpha=3,4$ separately and splitting the congruences modulo $24$ to congruences modulo $3$ and $8$ , one can verify that this system is solvable. We omit the proof of the case $p=2$ , $v_{2}(N)=1,2$ . If $p=3$ and $v_{3}(N)=\alpha\geq 2$ , then (38) is equivalent to

\begin{dcases}\sum_{1\leq\beta\leq\alpha}(3^{\beta}-1)r^{(3)}_{3^{\beta}}\equiv(3^{\alpha}-1)c_{3^{\alpha}}+(3^{\alpha-1}-1)c_{3^{\alpha-1}}\pmod{24}\\ \sum_{1\leq\beta\leq\alpha}(N3^{-\beta}-N)r^{(3)}_{3^{\beta}}\equiv(N3^{-\alpha}-N)c_{3^{\alpha}}+(N3^{-\alpha+1}-N)c_{3^{\alpha-1}}\pmod{24}\\ \sum_{\begin{subarray}{c}{\beta\in\{\alpha-1,\alpha\}}\\ {c_{3^{\beta}}\equiv r^{(3)}_{3^{\beta}}+1\bmod 2}\end{subarray}}\beta\equiv\sum_{1\leq\beta<\alpha-1,\,r^{(3)}_{3^{\beta}}\equiv 1\bmod 2}\beta\pmod{2}.\end{dcases}

As in the above case, by considering the subcases $2\mid\alpha$ and $2\nmid\alpha$ separately and splitting the congruences modulo $24$ to congruences modulo $3$ and $8$ , one can verify that this system is solvable. The case $p=3$ , $v_{3}(N)=1$ is obvious. Thereby we have shown (38) which concludes the proof. ∎

To state the main theorem of this section, let us introduce the system of congruences

(39)

\begin{dcases}\sum_{n\in\mathscr{B}_{N}}(n-1)c_{n}\equiv 0\pmod{24}\\ \sum_{n\in\mathscr{B}_{N}}(Nn^{-1}-N)c_{n}\equiv 0\pmod{24}\\ c_{p}\equiv 0\pmod{2},\,\forall p\mid N,p\neq 2,3\\ \sum_{2^{\beta}\in\mathscr{B}_{N}\colon c_{2^{\beta}}\equiv 1\bmod 2}\beta\equiv 0\pmod{2},\quad\sum_{3^{\beta}\in\mathscr{B}_{N}\colon c_{3^{\beta}}\equiv 1\bmod 2}\beta\equiv 0\pmod{2}.\end{dcases}

Then the element (37) is the zero in $\mathscr{S}(N)$ if and only if $(c_{n})_{n\in\mathscr{B}_{N}}$ satisfies (39) according to the definition of $Z(N)$ .

Definition 6.7.

Let $N\in\mathbb{Z}_{\geq 2}$ . For any ordering $b_{1},b_{2},\dots,b_{t}$ of $\mathscr{B}_{N}$ (which means $\mathscr{B}_{N}=\{b_{1},b_{2},\dots,b_{t}\}$ and $b_{i}\neq b_{j}$ for $i\neq j$ ), we define a sequence of positive integers $\Delta_{1},\Delta_{2},\dots,\Delta_{t}$ as follows:

\Delta_{i}=\min\{1\leq m\leq 24\colon\exists(c_{n})_{n\in\mathscr{B}_{N}}\text{ s.t. \eqref{eq:congruencesZN} holds and }c_{b_{j}}=0\,(j<i),\,c_{b_{i}}=m\}.

Note that $\Delta_{i}$ is well defined since the set in the right-hand side always contains $24$ . In addition, be careful that the definition does not imply $\sum_{1\leq j\leq t}\Delta_{j}e_{j}-\left(\sum_{1\leq j\leq t}\Delta_{j}\right)e_{1}\in Z(N)$ unless $\lvert\mathscr{B}_{N}\rvert=1$ .

Theorem 6.8.

With the notation of Definition 6.7, a complete system of representatives of $\mathscr{S}(N)$ is given by

(40)

\sum_{1\leq j\leq t}c_{b_{j}}e_{b_{j}}-\left(\sum_{1\leq j\leq t}c_{b_{j}}\right)e_{1},\quad 0\leq c_{b_{j}}<\Delta_{j}.

Proof.

Let $\mathbf{r}+Z(N)\in\mathscr{S}(N)$ be arbitrary. According to Lemma 6.6 we can assume that $\mathbf{r}=\sum_{n\in\mathscr{B}_{N}}r_{n}e_{n}-\left(\sum_{n\in\mathscr{B}_{N}}r_{n}\right)e_{1}$ with $r_{n}\in\mathbb{Z}$ . By the definition of $\Delta_{1}$ , there exists $\mathbf{d}_{1}=\sum_{n\in\mathscr{B}_{N}}d_{n}e_{n}-\left(\sum_{n\in\mathscr{B}_{N}}d_{n}\right)e_{1}\in Z(N)$ with $d_{b_{1}}=\Delta_{1}$ . Hence subtracting a multiple of $\mathbf{d}_{1}$ from $\mathbf{r}$ we may assume that $0\leq r_{b_{1}}<\Delta_{1}$ . Inductively, suppose we have adjusted $\mathbf{r}$ such that $0\leq r_{b_{j}}<\Delta_{j}$ for $j=1,2,\dots,J$ . Let $\mathbf{d}_{J+1}=\sum_{n\in\mathscr{B}_{N}}d_{n}e_{n}-\left(\sum_{n\in\mathscr{B}_{N}}d_{n}\right)e_{1}\in Z(N)$ with $d_{b_{j}}=0$ ( $j\leq J$ ) and $d_{b_{J+1}}=\Delta_{J+1}$ . This exists by the definition of $\Delta_{J+1}$ . Subtracting a multiple of $\mathbf{d}_{J+1}$ from $\mathbf{r}$ we can assume that $0\leq r_{b_{j}}<\Delta_{j}$ for $j=1,2,\dots,J+1$ . Therefore by induction $\mathbf{r}$ takes the form of (40) modulo $Z(N)$ . We have proved that each coset in $\mathscr{S}(N)$ can be represented by an element in (40).

Let $\mathbf{r}_{1}=\sum_{1\leq j\leq t}c_{b_{j}}e_{b_{j}}-\left(\sum_{1\leq j\leq t}c_{b_{j}}\right)e_{1}$ and $\mathbf{r}_{2}=\sum_{1\leq j\leq t}d_{b_{j}}e_{b_{j}}-\left(\sum_{1\leq j\leq t}d_{b_{j}}\right)e_{1}$ be two different representatives of the form (40). Let $1\leq J\leq t$ satisfy $c_{b_{j}}=d_{b_{j}}$ for $j<J$ and $c_{b_{J}}\neq d_{b_{J}}$ . Without loss of generality assume that $c_{b_{J}}<d_{b_{J}}$ . It follows that $0<d_{b_{J}}-c_{b_{J}}<\Delta_{J}$ . Then $\mathbf{r}_{2}-\mathbf{r}_{1}\not\in Z(N)$ by the definition of $\Delta_{J}$ . We have proved that the vectors in (40) are pairwise inequivalent modulo $Z(N)$ , which concludes the whole proof. ∎

Corollary 6.9.

Let $N\in\mathbb{Z}_{\geq 1}$ and $k\in\frac{1}{2}\mathbb{Z}$ . Then the characters (6) of eta-quotients of level $N$ , weight $k$ and cover index $2$ are in one-to-one correspondence with the vectors

\sum_{1\leq j\leq t}r_{b_{j}}e_{b_{j}}+\left(2k-\sum_{1\leq j\leq t}r_{b_{j}}\right)e_{1},\quad 0\leq r_{b_{j}}<\Delta_{j}.

Proof.

According to the discussion followed by Remark 6.3, the characters of eta-quotients of level $N$ , weight $k$ and cover index $2$ can be regarded as the set $\mathscr{S}(N,k)$ and the map $\mathscr{S}(N)\to\mathscr{S}(N,k)$ , $\mathbf{r}+Z(N)\mapsto\mathbf{r}+2ke_{1}+Z(N)$ is a bijection. The corollary follows from this and Theorem 6.8. ∎

In the remainder of this section, we give examples of concrete sequences $(\Delta_{i})_{1\leq i\leq t}$ ( $t=\lvert\mathscr{B}_{N}\rvert$ ) for some special $N$ .

Example 6.10.

Let $N=p^{\alpha}$ where $p\geq 5$ is a prime and $\alpha\in\mathbb{Z}_{\geq 1}$ . Then $\mathscr{B}_{N}=\{p\}$ and the sequence $(\Delta_{i})_{1\leq i\leq t}$ ( $t=\lvert\mathscr{B}_{N}\rvert$ ) contains a single element $\Delta_{1}$ . The system of congruences (39) is equivalent to $(p-1)c_{p}\equiv 0\bmod{24}$ and $c_{p}\equiv 0\bmod{2}$ in this case. Thus, $\Delta_{1}=\frac{24}{(12,p-1)}$ and a complete system of representatives of $\mathscr{S}(p^{\alpha})$ is given by

r_{p}(e_{p}-e_{1}),\quad 0\leq r_{p}<\frac{24}{(12,p-1)}.

As a consequence, for any $k\in\frac{1}{2}\mathbb{Z}$ there are exactly $\frac{24}{(12,p-1)}$ characters $\chi\colon\widetilde{\Gamma_{0}(p^{\alpha})}\to\mathbb{C}^{\times}$ of eta-quotients of level $p^{\alpha}$ , weight $k$ and cover index $2$ . It should be noted that, the character of an eta-quotient, say, of cover index $4$ may also be regarded as a character on $\widetilde{\Gamma_{0}(p^{\alpha})}$ (cf. the third fact of Remark 2.2). For instance, the character $\chi\colon\widetilde{\Gamma_{0}(p^{\alpha})^{8}}\to\mathbb{C}^{\times}$ of $\eta(\tau)^{1/4}\eta(p\tau)^{3/4}$ descends to a character on $\widetilde{\Gamma_{0}(p^{\alpha})}=\widetilde{\Gamma_{0}(p^{\alpha})^{2}}$ . Such characters are not in the list above.

Example 6.11.

Let $N=2^{\alpha}$ with $\alpha\geq 3$ . Then $\mathscr{B}_{N}=\{2^{\alpha-2},2^{\alpha-1},2^{\alpha}\}$ which is arranged in the increasing order. We need to calculate the sequence $\Delta_{1},\Delta_{2},\Delta_{3}$ . The system of congruences (39) is equivalent to

\begin{dcases}(2^{\alpha-2}-1)c_{2^{\alpha-2}}+(2^{\alpha-1}-1)c_{2^{\alpha-1}}+(2^{\alpha}-1)c_{2^{\alpha}}\equiv 0\pmod{24}\\ (4-2^{\alpha})c_{2^{\alpha-2}}+(2-2^{\alpha})c_{2^{\alpha-1}}+(1-2^{\alpha})c_{2^{\alpha}}\equiv 0\pmod{24}\\ \sum_{\begin{subarray}{c}{\beta\in\{\alpha-2,\alpha-1,\alpha\}}\\ {c_{2^{\beta}}\equiv 1\bmod 2}\end{subarray}}\beta\equiv 0\pmod{2}.\end{dcases}

One can verify directly that

(\Delta_{1},\Delta_{2},\Delta_{3})=\begin{dcases}(2,24,8)&\text{ if }\alpha\geq 3\text{ and }2\mid\alpha\\ (2,8,24)&\text{ if }\alpha\geq 3\text{ and }2\nmid\alpha.\end{dcases}

Therefore, a complete system of representatives of $\mathscr{S}(2^{\alpha})$ with $\alpha\geq 3$ is given by

r_{2^{\alpha-2}}e_{2^{\alpha-2}}+r_{2^{\alpha-1}}e_{2^{\alpha-1}}+r_{2^{\alpha}}e_{2^{\alpha}}-(r_{2^{\alpha-2}}+r_{2^{\alpha-1}}+r_{2^{\alpha}})e_{1},

where $r_{\alpha-2}\in\{0,1\}$ and $0\leq r_{2^{\alpha-1}}<24$ , $0\leq r_{2^{\alpha}}<8$ if $2\mid\alpha$ ; $0\leq r_{2^{\alpha-1}}<8$ , $0\leq r_{2^{\alpha}}<24$ if $2\nmid\alpha$ . As a consequence, for any $k\in\frac{1}{2}\mathbb{Z}$ there are exactly $384$ characters $\chi\colon\widetilde{\Gamma_{0}(2^{\alpha})}\to\mathbb{C}^{\times}$ of eta-quotients of level $2^{\alpha}$ , weight $k$ and cover index $2$ when $\alpha\geq 3$ . In addition, it is immediate that for $N=2^{2}$ we have $\mathscr{B}_{4}=\{2^{1},2^{2}\}$ and $(\Delta_{1},\Delta_{2})=(24,8)$ and for $N=2^{1}$ we have $\mathscr{B}_{2}=\{2^{1}\}$ and $\Delta_{1}=24$ . Thus, there are exactly $192$ ( $24$ respectively) characters of eta-quotients of level $4$ ( $2$ respectively), cover index $2$ and any fixed weight $\in\frac{1}{2}\mathbb{Z}$ .

Example 6.12.

Let $N=3^{\alpha}$ with $\alpha\geq 2$ . Then $\mathscr{B}_{N}=\{3^{\alpha-1},3^{\alpha}\}$ which is arranged in the increasing order. We need to calculate the sequence $\Delta_{1},\Delta_{2}$ . The system of congruences (39) is equivalent to

\begin{dcases}(3^{\alpha-1}-1)c_{3^{\alpha-1}}+(3^{\alpha}-1)c_{3^{\alpha}}\equiv 0\pmod{24}\\ (3-3^{\alpha})c_{3^{\alpha-1}}+(1-3^{\alpha})c_{3^{\alpha}}\equiv 0\pmod{24}\\ \sum_{\begin{subarray}{c}{\beta\in\{\alpha-1,\alpha\}}\\ {c_{3^{\beta}}\equiv 1\bmod 2}\end{subarray}}\beta\equiv 0\pmod{2}.\end{dcases}

One can verify directly that

(\Delta_{1},\Delta_{2})=\begin{dcases}(12,3)&\text{ if }\alpha\geq 2\text{ and }2\mid\alpha\\ (3,12)&\text{ if }\alpha\geq 2\text{ and }2\nmid\alpha.\end{dcases}

Therefore, a complete system of representatives of $\mathscr{S}(3^{\alpha})$ with $\alpha\geq 2$ is given by

r_{3^{\alpha-1}}e_{3^{\alpha-1}}+r_{3^{\alpha}}e_{3^{\alpha}}-(r_{3^{\alpha-1}}+r_{3^{\alpha}})e_{1},

where $0\leq r_{3^{\alpha-1}}<12$ , $0\leq r_{3^{\alpha}}<3$ if $2\mid\alpha$ ; $0\leq r_{3^{\alpha-1}}<3$ , $0\leq r_{3^{\alpha}}<12$ if $2\nmid\alpha$ . As a consequence, for any $k\in\frac{1}{2}\mathbb{Z}$ there are exactly $36$ characters $\chi\colon\widetilde{\Gamma_{0}(3^{\alpha})}\to\mathbb{C}^{\times}$ of eta-quotients of level $3^{\alpha}$ , weight $k$ and cover index $2$ when $\alpha\geq 2$ . In addition, it is direct that for $N=3^{1}$ we have $\mathscr{B}_{3}=\{3^{1}\}$ and $\Delta_{1}=12$ . Thus, there are exactly $12$ characters of eta-quotients of level $3$ , cover index $2$ and any fixed weight $\in\frac{1}{2}\mathbb{Z}$ .

Example 6.13.

Let $N=4p^{\alpha}$ where $p\geq 5$ is a prime and $\alpha\in\mathbb{Z}_{\geq 1}$ . Then $\mathscr{B}_{N}=\{2^{1},2^{2},p\}$ which is arranged in the increasing order. We need to calculate the sequence $\Delta_{1},\Delta_{2},\Delta_{3}$ . The system of congruences (39) is equivalent to

\begin{dcases}c_{2}+3c_{4}+(p-1)c_{p}\equiv 0\pmod{24}\\ -2p^{\alpha}c_{2}-3p^{\alpha}c_{4}+(4p^{\alpha-1}-4p^{\alpha})c_{p}\equiv 0\pmod{24}\\ c_{2},\,c_{p}\equiv 0\pmod{2}.\end{dcases}

Using elementary number theory one can show that

\Delta_{1}=2\cdot(12,p-1),\quad\Delta_{2}=8,\quad\Delta_{3}=\frac{24}{(12,p-1)}.

Therefore, a complete system of representatives of $\mathscr{S}(4p^{\alpha})$ with $p\geq 5$ and $\alpha\geq 1$ is given by

r_{2}e_{2}+r_{4}e_{4}+r_{p}e_{p}-(r_{2}+r_{4}+r_{p})e_{1}

with

0\leq r_{2}<2\cdot(12,p-1),\quad 0\leq r_{4}<8,\quad 0\leq r_{p}<\frac{24}{(12,p-1)}.

As a consequence, for any $k\in\frac{1}{2}\mathbb{Z}$ there are exactly $384$ characters $\chi\colon\widetilde{\Gamma_{0}(4p^{\alpha})}\to\mathbb{C}^{\times}$ of eta-quotients of level $4p^{\alpha}$ , weight $k$ and cover index $2$ when $p\geq 5$ and $\alpha\geq 1$ .

Example 6.14.

Let $N=p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}}$ where $p_{1},p_{2}\geq 5$ are distinct primes and $\alpha_{1},\alpha_{2}\in\mathbb{Z}_{\geq 1}$ . Then $\mathscr{B}_{N}=\{p_{1},p_{2}\}$ . We need to calculate the sequence $\Delta_{1},\Delta_{2}$ . The system of congruences (39) is equivalent to

\begin{dcases}(p_{1}-1)c_{p_{1}}+(p_{2}-1)c_{p_{2}}\equiv 0\pmod{24}\\ (p_{1}^{\alpha_{1}-1}p_{2}^{\alpha_{2}}-p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}})c_{p_{1}}+(p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}-1}-p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}})c_{p_{2}}\equiv 0\pmod{24}\\ c_{p_{1}},\,c_{p_{2}}\equiv 0\pmod{2}.\end{dcases}

Using elementary number theory one can show that

	$\displaystyle\Delta_{1}$	$\displaystyle=\frac{24\cdot(12,p_{2}-1)}{(12,(p_{1}-1)(p_{2}-1))}\cdot\left(\frac{12\cdot(12,p_{2}-1)}{(12,(p_{1}-1)(p_{2}-1))},p_{1}-p_{2}\right)^{-1},$
	$\displaystyle\Delta_{2}$	$\displaystyle=\frac{24}{(12,p_{2}-1)}.$

The expression for $\Delta_{1}$ seems to be complicated but it actually depends only on $p_{1}\bmod{12}$ and $p_{2}\bmod{12}$ . For instance, if $p_{2}\equiv 1\bmod{12}$ , then $\Delta_{1}$ is simplified to $\frac{24}{(12,p_{1}-1)}$ . A complete system of representatives of $\mathscr{S}(p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}})$ is given by

r_{p_{1}}e_{p_{1}}+r_{p_{2}}e_{p_{2}}-(r_{p_{1}}+r_{p_{2}})e_{1}

with

0\leq r_{p_{1}}<\Delta_{1},\quad 0\leq r_{p_{2}}<\Delta_{2}.

As a consequence, for any $k\in\frac{1}{2}\mathbb{Z}$ there are exactly $\Delta_{1}\Delta_{2}$ characters $\chi\colon\widetilde{\Gamma_{0}(p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}})}\to\mathbb{C}^{\times}$ of eta-quotients of level $p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}}$ , weight $k$ and cover index $2$ when $p_{1},p_{2}\geq 5$ are distinct primes and $\alpha_{1},\alpha_{2}\geq 1$ .

7. Dimension formulas for weights $1/2$ , $1$ , $3/2$ and small levels

In this section, we give values of all $\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(N),\chi)$ that can be calculated by Theorem 4.2 (throughout this section, $t$ in this theorem always takes the value $0$ ) for $k=1/2$ , $1$ and $3/2$ , $N\in\mathbb{Z}_{\geq 1}$ and $\chi$ being the character of an eta-quotient of level $N$ , weight $k$ and cover index $2$ . There are only finitely many such formulas since the right-hand side of (21) tends to $2$ when $N\to+\infty$ and the set of $\chi$ is finite for each $N$ and $k$ (cf. Corollary 6.9). The following lemma gives an upper bound of $N$ that should be considered.

Lemma 7.1.

Let $N$ be a positive integer and let $m$ , $\varepsilon_{2}$ , $\varepsilon_{3}$ and $\varepsilon_{\infty}$ be defined in (9), (10), (11) and (12) respectively. If $N\geq 1024$ , we have

\varepsilon_{2}<\frac{m}{96},\quad\varepsilon_{3}<\frac{m}{128},\quad\varepsilon_{\infty}\leq\frac{m}{32}.

Consequently, in Theorem 4.2, if $k\leq 3/2$ , $t=0$ and $N\geq 1024$ , then (21) does not hold.

For $\varepsilon_{\infty}$ , a stronger estimate⁸⁸8This is the best estimate if we want it to hold for all $N$ since it becomes equality when $N$ is a square. is $\varepsilon_{\infty}\leq m\cdot N^{-1/2}$ with $N\in\mathbb{Z}_{\geq 1}$ which we will prove below.

Proof.

Suppose $N=\prod_{1\leq j\leq s}p_{j}^{\alpha_{j}}$ where $p_{1}<p_{2}<\dots<p_{s}$ are primes, $s\in\mathbb{Z}_{\geq 0}$ and $\alpha_{j}\in\mathbb{Z}_{\geq 1}$ . Note that for $p=p_{j}$ , $\alpha=\alpha_{j}$ we have

\frac{p^{[\alpha/2]}+p^{[\alpha/2-1/2]}}{p^{\alpha}+p^{\alpha-1}}=\begin{dcases}\frac{2}{p^{\alpha/2}(p^{1/2}+p^{-1/2})}&\text{ if }2\nmid\alpha,\\ \frac{1}{p^{\alpha/2}}&\text{ if }2\mid\alpha.\end{dcases}

Thereby, according to [29, Corollary 6.3.24(b)]

(41)

\frac{\varepsilon_{\infty}}{m}=\prod_{1\leq j\leq s}\frac{p_{j}^{[\alpha_{j}/2]}+p_{j}^{[\alpha_{j}/2-1/2]}}{p_{j}^{\alpha_{j}}+p_{j}^{\alpha_{j}-1}}\leq\prod_{1\leq j\leq s}\frac{1}{p_{j}^{\alpha_{j}/2}}=N^{-1/2}.

Now we begin to prove the inequalities for $\varepsilon_{2}$ and $\varepsilon_{3}$ , so suppose $N\geq 1024$ . If $s=1,2,3$ , then $\varepsilon_{2}/m\leq 2^{3}/1025<1/96$ , $\varepsilon_{3}/m\leq 2^{3}/1025<1/128$ . If $s\geq 4$ , we consider two subcases: $\varepsilon_{2}=0$ and $\varepsilon_{2}\neq 0$ . If $\varepsilon_{2}=0$ , obviously $\varepsilon_{2}/m<1/96$ . If $\varepsilon_{2}\neq 0$ , then $\genfrac{(}{)}{}{}{-4}{p_{j}}\neq-1$ and hence $p_{j}=2$ or $p_{j}\equiv 1\bmod 4$ for $1\leq j\leq s$ . If $p_{1}\neq 2$ , then $p_{1}^{\alpha_{1}}\geq 5$ , $p_{2}^{\alpha_{2}}\geq 13$ and $p_{j}^{\alpha_{j}}\geq 17$ for $j\geq 3$ . Therefore

\frac{\varepsilon_{2}}{m}\leq\frac{2}{5+1}\cdot\frac{2}{13+1}\cdot\left(\frac{2}{17+1}\right)^{s-2}<\frac{1}{96}.

If $p_{1}=2$ , then $p_{2}^{\alpha_{2}}\geq 5$ , $p_{3}^{\alpha_{3}}\geq 13$ and $p_{j}^{\alpha_{j}}\geq 17$ for $j\geq 4$ . Therefore

\frac{\varepsilon_{2}}{m}\leq\frac{1}{2+1}\cdot\frac{2}{5+1}\cdot\frac{2}{13+1}\cdot\left(\frac{2}{17+1}\right)^{s-3}<\frac{1}{96}.

This proves the inequality for $\varepsilon_{2}$ . To prove the inequality for $\varepsilon_{3}$ when $s\geq 4$ , we consider the subcases $\varepsilon_{3}=0$ and $\varepsilon_{3}\neq 0$ separately and proceed as in the proof for $\varepsilon_{2}$ .

Finally, with the notation of Theorem 4.2 suppose that $k\leq 3/2$ , $t=0$ and $N\geq 1024$ . Assume by contradiction that (21) holds, then

\frac{3}{2}\geq k>2-\frac{6}{m}\varepsilon_{2}-\frac{8}{m}\varepsilon_{3}-\frac{12}{m}\varepsilon_{\infty}>2-\frac{6}{96}-\frac{8}{128}-\frac{12}{32}

which is absurd. Hence (21) does not hold. ∎

The algorithm for presenting all $\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(N),\chi)$ described above is as follows:

(a)

Let $k$ range over $\{1/2,1,3/2\}$ and let $N$ range over the positive integers $\leq 1023$ . We do not miss anything according to Lemma 7.1.
(b)

Given $k,N$ , we arrange $\mathscr{B}_{N}$ in any order and calculate the sequence $\Delta_{1},\dots,\Delta_{t}$ (cf. Definition 6.7).
(c)

Let $\mathbf{r}=(r_{n})_{n\mid N}$ range over the vectors described in Corollary 6.9. According to this corollary, the corresponding $\chi_{\mathbf{r}}$ ranges over exactly all characters of eta-quotients of level $N$ , weight $k$ and cover index $2$ .
(d)

Given $k,N$ and $\mathbf{r}$ , we calculate $\mathbf{x}=(x_{c})_{c\mid N}$ via (19).
(e)

We determine whether (21) holds. If it holds, then we calculate $\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(N),\chi_{\mathbf{r}})$ by (22) where $t=0$ . If it does not hold, $\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(N),\chi_{\mathbf{r}})$ can not be calculated by Theorem 4.2. Nevertheless we can obtain an upper bound and a lower bound of $\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(N),\chi_{\mathbf{r}})$ (cf. Remark 4.3).

Remark 7.2.

For a fixed $k$ , one can shrink the range of $N$ by picking out those $N\leq 1023$ such that $k>2-\frac{6}{m}\varepsilon_{2}-\frac{8}{m}\varepsilon_{3}-\frac{12}{m}\varepsilon_{\infty}$ . Hence, for $k=1/2$ , $N$ need only to range over $\{1\leq N\leq 21\}\cup\{24,25,27,32,36,49,50\}$ ; for $k=1$ , $N$ need only to range over $\{1\leq N\leq 22\}\cup\{24\leq N\leq 32\}\cup\{34,36,37,39,40,45,48,49,50,54,64,72,75,81,98,100,121,169\}$ ; for $k=3/2$ , $N$ need only to range over a subset of $\{1\leq N\leq 529\}$ .

All the above steps can be easily implemented in any computer algebra system. We program them using SageMath. See Appendix A for the usage of the code.

7.1. The weight $1/2$

The dimensions obtained by the above algorithm are summarized in Table 1.

Table 1. For each

N

a

means the total number of characters (cf. Corollary 6.9) and

v

means the number of characters

\chi

such that

\dim_{\mathbb{C}}M_{1/2}(\Gamma_{0}(N),\chi)

can be computed using Theorem 4.2,

d_{j}

means the number of spaces of dimension

j

for

j=0,1,2

$N$	$a$	$v$	$d_{0}$	$d_{1}$	$d_{2}$	$N$	$a$	$v$	$d_{0}$	$d_{1}$	$d_{2}$
$1$	$1$	$1$	$0$	$1$	$0$	$2$	$24$	$24$	$20$	$4$	$0$
$3$	$12$	$12$	$10$	$2$	$0$	$4$	$192$	$146$	$136$	$10$	$0$
$5$	$6$	$6$	$4$	$2$	$0$	$6$	$288$	$160$	$148$	$12$	$0$
$7$	$4$	$4$	$2$	$2$	$0$	$8$	$384$	$214$	$198$	$16$	$0$
$9$	$36$	$18$	$15$	$3$	$0$	$10$	$48$	$32$	$24$	$8$	$0$
$12$	$1152$	$318$	$286$	$32$	$0$	$13$	$2$	$2$	$0$	$2$	$0$
$16$	$384$	$127$	$108$	$18$	$1$	$18$	$288$	$56$	$42$	$12$	$2$
$20$	$384$	$6$	$0$	$6$	$0$	$24$	$2304$	$24$	$0$	$24$	$0$
$25$	$6$	$3$	$1$	$1$	$1$	$27$	$36$	$2$	$0$	$2$	$0$
$32$	$384$	$10$	$0$	$8$	$2$	$36$	$1152$	$12$	$0$	$6$	$6$
$49$	$4$	$1$	$0$	$0$	$1$	$50$	$48$	$4$	$0$	$0$	$4$

Example 7.3.

Let us consider the case $N=50$ . We have $\mathscr{B}_{50}=\{2,5\}$ and $(\Delta_{1},\Delta_{2})=(8,6)$ as one can verify directly. Thus, there are totally $48$ characters of eta-quotients of level $50$ , weight $1/2$ and cover index $2$ . They are exactly the characters on $\widetilde{\Gamma_{0}(50)^{2}}$ of the following eta-quotients:

\eta(\tau)^{1-r_{2}-r_{5}}\eta(2\tau)^{r_{2}}\eta(5\tau)^{r_{5}},\quad 0\leq r_{2}<8,\,0\leq r_{5}<6.

Among the $48$ characters, for precisely the four characters $\chi$ given by

(42)

(r_{2},r_{5})=(0,0),\,(1,0),\,(2,0),\,(7,4)

one can compute $\dim_{\mathbb{C}}M_{1/2}(\Gamma_{0}(50),\chi)$ using Theorem 4.2 (that is, (21) holds with $t=0$ ) and the four dimensions all equal $2$ . For the other $44$ characters $\chi$ our method only gives that $0\leq\dim_{\mathbb{C}}M_{1/2}(\Gamma_{0}(50),\chi)\leq 4$ (cf. Remark 4.3). Let $\chi_{1},\chi_{2},\chi_{3},\chi_{4}$ denote respectively the characters given by (42). Then according to (14) and Lemma 6.1 we have

	$\displaystyle M_{1/2}(\Gamma_{0}(50),\chi_{1})$	$\displaystyle=\mathbb{C}\eta(\tau)\oplus\mathbb{C}\eta(25\tau),$
	$\displaystyle M_{1/2}(\Gamma_{0}(50),\chi_{2})$	$\displaystyle=\mathbb{C}\eta(2\tau)\oplus\mathbb{C}\eta(50\tau),$
	$\displaystyle M_{1/2}(\Gamma_{0}(50),\chi_{3})$	$\displaystyle=\mathbb{C}\eta(\tau)^{-1}\eta(2\tau)^{2}\oplus\mathbb{C}\eta(25\tau)^{-1}\eta(50\tau)^{2}.$

Finding explicit generators of $M_{1/2}(\Gamma_{0}(50),\chi_{4})$ seems to be a bit harder since $\eta(\tau)^{-10}\eta(2\tau)^{7}\eta(5\tau)^{4}$ is only a weakly holomorphic modular form. However, we have $7e_{2}+4e_{5}-11e_{1}\sim-e_{2}+e_{1}$ module $Z(50)$ which can be verified by (39). It follows that

M_{1/2}(\Gamma_{0}(50),\chi_{4})=\mathbb{C}\eta(\tau)^{2}\eta(2\tau)^{-1}\oplus\mathbb{C}\eta(25\tau)^{2}\eta(50\tau)^{-1}.

Finally note that $M_{1/2}(\Gamma_{0}(50),\chi_{j})=S_{1/2}(\Gamma_{0}(50),\chi_{j})$ for $j=1,2$ , while the Eisenstein subspace of $M_{1/2}(\Gamma_{0}(50),\chi_{j})$ is nontrivial for $j=3,4$ .

Example 7.4.

We consider another example $N=20$ in which there are totally $384$ characters. They have been described in Example 6.13 with $p=5$ , $\alpha=1$ . We have $\mathscr{B}_{20}=\{2,4,5\}$ and $(\Delta_{1},\Delta_{2},\Delta_{3})=(8,8,6)$ . Therefore the characters are exactly those on $\widetilde{\Gamma_{0}(20)^{2}}$ of the following eta-quotients:

\eta(\tau)^{1-r_{2}-r_{4}-r_{5}}\eta(2\tau)^{r_{2}}\eta(4\tau)^{r_{4}}\eta(5\tau)^{r_{5}},\quad 0\leq r_{2}<8,\,0\leq r_{4}<8,\,0\leq r_{5}<6.

Among the $384$ characters, for precisely the six characters $\chi$ given by

(43)

(r_{2},r_{4},r_{5})=(1,6,1),\,(3,0,5),\,(3,2,5),\,(5,6,0),\,(7,0,4),\,(7,2,4)

one can compute $\dim_{\mathbb{C}}M_{1/2}(\Gamma_{0}(20),\chi)$ using Theorem 4.2 and the six dimensions all equal $1$ . For the other $378$ characters $\chi$ our method only gives that $0\leq\dim_{\mathbb{C}}M_{1/2}(\Gamma_{0}(20),\chi)\leq 2$ (cf. Remark 4.3). Let $\chi_{1},\chi_{2},\chi_{3},\chi_{4},\chi_{5},\chi_{6}$ denote respectively the characters given by (43). One can verify using (14) and Lemma 6.1 that

$\displaystyle M_{1/2}(\Gamma_{0}(20),\chi_{1})$	$\displaystyle=\mathbb{C}\eta(5\tau)^{-2}\eta(10\tau)^{5}\eta(20\tau)^{-2},\quad$	$\displaystyle M_{1/2}(\Gamma_{0}(20),\chi_{2})$	$\displaystyle=\mathbb{C}\eta(5\tau)^{2}\eta(10\tau)^{-1},$
$\displaystyle M_{1/2}(\Gamma_{0}(20),\chi_{3})$	$\displaystyle=\mathbb{C}\eta(10\tau)^{-1}\eta(20\tau)^{2},\quad$	$\displaystyle M_{1/2}(\Gamma_{0}(20),\chi_{4})$	$\displaystyle=\mathbb{C}\eta(\tau)^{-2}\eta(2\tau)^{5}\eta(4\tau)^{-2},$
$\displaystyle M_{1/2}(\Gamma_{0}(20),\chi_{5})$	$\displaystyle=\mathbb{C}\eta(\tau)^{2}\eta(2\tau)^{-1},\quad$	$\displaystyle M_{1/2}(\Gamma_{0}(20),\chi_{6})$	$\displaystyle=\mathbb{C}\eta(2\tau)^{-1}\eta(4\tau)^{2}.$

Remark 7.5.

There are totally $188$ nonzero spaces $M_{1/2}(\Gamma_{0}(N),\chi)$ whose dimensions can be computed by Theorem 4.2 according to Table 1. Note that we restrict ourselves to the characters $\chi$ of eta-quotients of cover index $2$ , so the dimension formulas summarized in Table 1 are not direct consequences of the well known theorem of Serre and Stark [14] that gives an explicit basis of $M_{1/2}(\Gamma_{0}(N),\chi_{1}\chi)$ where $4\mid N$ , $\chi_{1}$ is the character of Euler theta function $\theta(\tau)=\eta(\tau)^{-2}\eta(2\tau)^{5}\eta(4\tau)^{-2}$ and $\chi$ is the character induced by any even Dirichlet character $(\mathbb{Z}/N\mathbb{Z})^{\times}\to\mathbb{C}^{\times}$ . For instance, when $N=12$ , there are two even Dirichlet characters and hence the theorem of Serre and Stark gives explicit bases of two spaces $M_{1/2}(\Gamma_{0}(12),\chi)$ . As a comparison, there are totally $1152$ characters of eta-quotients of weight $1/2$ , level $12$ and cover index $2$ among which the dimensions of $318$ spaces can be computed by Theorem 4.2— $286$ spaces are zero-dimensional and $32$ spaces are one-dimensional.

Remark 7.6.

Zagier observed that there are exactly $14$ primitive (which means the greatest common divisor of all the $n$ with $r_{n}\neq 0$ is equal to $1$ ) eta-quotients of integral exponents that are modular forms of weight $1/2$ . This and the similar conjectures on higher weights were proved by Mersmann in his Master’s thesis. See [21] for a simpler proof. It seems that all the generators of the $188$ nonzero spaces $M_{1/2}(\Gamma_{0}(N),\chi)$ whose dimensions can be computed by Theorem 4.2 (cf. Table 1) take the form $f(m\tau)$ where $f$ is one of the $14$ primitive eta-quotients and $m$ is a suitable positive integer. This is verified by the SageMath program for $N=4,20,50$ . See Appendix A for how to find the code doing this.

7.2. The weight $1$

The dimensions obtained by the algorithm described above are summarized in Table LABEL:table:wt1. See Appendix A for the SageMath code that generates this kind of tables. Note that the characters are originally defined on $\widetilde{\Gamma_{0}(N)^{2}}$ since we are considering eta-quotients of cover index $2$ but they actually descend to characters on the matrix groups $\Gamma_{0}(N)$ since the weight is $1$ .

Table 2. For each

N

a

means the total number of characters (cf. Corollary 6.9) and

v

means the number of characters

\chi

such that

\dim_{\mathbb{C}}M_{1}(\Gamma_{0}(N),\chi)

can be computed using Theorem 4.2,

d_{j}

means the number of spaces of dimension

j

for

0\leq j\leq 8

$N$	$a$	$v$	$d_{0}$	$d_{1}$	$d_{2}$	$d_{3}$	$d_{4}$	$d_{5}$	$d_{6}$	$d_{7}$	$d_{8}$
$1$	$1$	$1$	$0$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
$2$	$24$	$24$	$17$	$7$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
$3$	$12$	$12$	$7$	$5$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
$4$	$192$	$173$	$142$	$31$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
$5$	$6$	$6$	$3$	$3$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
$6$	$288$	$250$	$189$	$60$	$1$	$0$	$0$	$0$	$0$	$0$	$0$
$7$	$4$	$4$	$1$	$3$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
$8$	$384$	$331$	$247$	$83$	$1$	$0$	$0$	$0$	$0$	$0$	$0$
$9$	$36$	$29$	$17$	$11$	$1$	$0$	$0$	$0$	$0$	$0$	$0$
$10$	$48$	$46$	$20$	$22$	$4$	$0$	$0$	$0$	$0$	$0$	$0$
$11$	$12$	$1$	$0$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
$12$	$1152$	$947$	$617$	$305$	$24$	$1$	$0$	$0$	$0$	$0$	$0$
$13$	$2$	$2$	$0$	$1$	$1$	$0$	$0$	$0$	$0$	$0$	$0$
$14$	$96$	$21$	$0$	$20$	$1$	$0$	$0$	$0$	$0$	$0$	$0$
$15$	$72$	$18$	$0$	$16$	$2$	$0$	$0$	$0$	$0$	$0$	$0$
$16$	$384$	$293$	$166$	$108$	$18$	$1$	$0$	$0$	$0$	$0$	$0$
$17$	$6$	$4$	$0$	$4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
$18$	$288$	$217$	$113$	$75$	$25$	$3$	$1$	$0$	$0$	$0$	$0$
$19$	$4$	$3$	$0$	$3$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
$20$	$384$	$96$	$0$	$72$	$24$	$0$	$0$	$0$	$0$	$0$	$0$
$21$	$24$	$14$	$0$	$8$	$6$	$0$	$0$	$0$	$0$	$0$	$0$
$22$	$96$	$1$	$0$	$0$	$1$	$0$	$0$	$0$	$0$	$0$	$0$
$24$	$2304$	$658$	$0$	$480$	$176$	$0$	$2$	$0$	$0$	$0$	$0$
$25$	$6$	$4$	$1$	$1$	$1$	$1$	$0$	$0$	$0$	$0$	$0$
$26$	$48$	$4$	$0$	$0$	$4$	$0$	$0$	$0$	$0$	$0$	$0$
$27$	$36$	$15$	$0$	$9$	$5$	$1$	$0$	$0$	$0$	$0$	$0$
$28$	$384$	$11$	$0$	$0$	$10$	$1$	$0$	$0$	$0$	$0$	$0$
$30$	$576$	$2$	$0$	$0$	$0$	$0$	$2$	$0$	$0$	$0$	$0$
$32$	$384$	$146$	$0$	$100$	$36$	$8$	$2$	$0$	$0$	$0$	$0$
$36$	$1152$	$412$	$0$	$198$	$168$	$30$	$15$	$0$	$1$	$0$	$0$
$37$	$2$	$1$	$0$	$0$	$1$	$0$	$0$	$0$	$0$	$0$	$0$
$40$	$768$	$2$	$0$	$0$	$0$	$0$	$2$	$0$	$0$	$0$	$0$
$45$	$72$	$2$	$0$	$0$	$0$	$0$	$2$	$0$	$0$	$0$	$0$
$48$	$2304$	$58$	$0$	$0$	$0$	$16$	$40$	$0$	$2$	$0$	$0$
$49$	$4$	$2$	$0$	$0$	$1$	$0$	$1$	$0$	$0$	$0$	$0$
$50$	$48$	$26$	$0$	$0$	$19$	$0$	$0$	$7$	$0$	$0$	$0$
$54$	$288$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$1$	$0$	$0$
$64$	$384$	$18$	$0$	$0$	$0$	$0$	$16$	$0$	$2$	$0$	$0$
$72$	$2304$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$2$
$75$	$72$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	$2$	$0$	$0$
$81$	$36$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$1$	$0$	$0$
$98$	$96$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$1$
$121$	$12$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$1$	$0$	$0$

We focus on the level $98$ in the rest of this subsection. According to Table LABEL:table:wt1, there are totally $96$ characters $\chi$ of eta-quotients of weight $1$ , level $98$ and cover index $2$ . They correspond to $96$ independent spaces $M_{1}(\Gamma_{0}(98),\chi)$ among which the dimension of exactly one space can be computed by Theorem 4.2, namely, the space where $\chi$ is the character of $\eta(\tau)^{-15}\eta(2\tau)^{16}\eta(7\tau)$ . For this $\chi$ we have $\dim_{\mathbb{C}}M_{1}(\Gamma_{0}(98),\chi)=8$ . One can show that $\chi\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right)=\genfrac{(}{)}{}{}{-7}{d}$ by, for instance, a result of Gordon, Hughes and Newman (cf. [39, Theorem 1.64]) or by checking directly for $\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right)$ being in a set of generators of $\Gamma_{0}(98)$ . Thus, an explicit basis can be given using Eisenstein series constructed by Weisinger [40]. See also [29, Theorem 8.5.23]. This shows there is no cusp form in $M_{1}(\Gamma_{0}(98),\chi)$ , which can also be proved using the tools developed in [15].

One may ask: is the space $M_{1}(\Gamma_{0}(98),\chi)$ generated by holomorphic eta-quotients? The answer is no. According to (14), the functions

	$\displaystyle\eta(\tau)^{-1}\eta(2\tau)^{2}\eta(7\tau)^{-1}\eta(14\tau)^{2}$	$\displaystyle=q+q^{2}+q^{4}+q^{7}+q^{8}+q^{9}+2q^{11}+q^{14}+\dots$
	$\displaystyle\eta(\tau)^{2}\eta(2\tau)^{-1}\eta(7\tau)^{2}\eta(14\tau)^{-1}$	$\displaystyle=1-2q+2q^{4}-2q^{7}+4q^{8}-2q^{9}-4q^{11}+6q^{16}+\dots$
	$\displaystyle\eta(7\tau)^{-1}\eta(14\tau)^{2}\eta(49\tau)^{-1}\eta(98\tau)^{2}$	$\displaystyle=q^{7}+q^{14}+q^{28}+q^{49}+q^{56}+q^{63}+2q^{77}+q^{98}+\dots$
	$\displaystyle\eta(7\tau)^{2}\eta(14\tau)^{-1}\eta(49\tau)^{2}\eta(98\tau)^{-1}$	$\displaystyle=1-2q^{7}+2q^{28}-2q^{49}+4q^{56}-2q^{63}-4q^{77}+6q^{112}+\dots$

are holomorphic eta-quotients of level $98$ . Using Lemma 6.1 the characters of these four modular forms are equal to $\chi$ and hence they belong to $M_{1}(\Gamma_{0}(98),\chi)$ . By their $q$ -coefficients these four functions are linearly independent. However there are no other holomorphic eta-quotients of cover index $2$ in $M_{1}(\Gamma_{0}(98),\chi)$ so this space can not be generated by a set consisting of all eta-quotients. This can be seen as follows. Let $f(\tau)=\prod_{n\mid 98}\eta(n\tau)^{r_{n}}$ be in $M_{1}(\Gamma_{0}(98),\chi)$ with $r_{n}\in\mathbb{Z}$ . Then $x_{c}\in\mathbb{Z}_{\geq 0}$ for any $c\mid 98$ where $x_{c}$ is given by (19). According to Theorem 3.2 (applied to $f$ ), (14) and the complete set of representatives of $\Gamma_{0}(98)\backslash\mathbb{P}^{1}(\mathbb{Q})$ described in the proof of Proposition 3.4 we have $x_{1}+x_{2}+6x_{7}+6x_{14}+x_{49}+x_{98}=336$ of which there are only finitely many solutions $(x_{c})_{c\mid 98}$ . For each solution, we obtain $(r_{n})_{n\mid 98}$ via (27) and then check whether $r_{n}\in\mathbb{Z}$ and the character corresponding to $(r_{n})_{n\mid 98}$ is the same as $\chi$ using Lemma 6.1. In this way we find that there are exactly four holomorphic eta-quotients of cover index $2$ in $M_{1}(\Gamma_{0}(98),\chi)$ (totally $69$ holomorphic eta-quotients, $57$ of which are primitive).

7.3. The weight $3/2$

There are totally $17862$ spaces $M_{3/2}(\Gamma_{0}(N),\chi)$ whose dimensions can be calculated using Theorem 4.2. Among these spaces, the largest level is $N=400$ and the largest dimension is $48$ . See Appendix A for the usage of SageMath code that gives these data. The following observation will be used later: the dimensions of these $17862$ spaces are all greater than $1$ when $N>36$ .

8. Application: one-dimensional spaces and Eisenstein series of rational weights

In [16, Section 7], the author gives infinitely many identities whose left-hand sides are Eisenstein series and right-hand sides are eta-quotients of rational weights for the levels $2$ and $3$ . Using Theorem 4.2 (with $t=0$ ), we find new identities of this kind for the level $4$ in this section.

First let us rewrite Theorem 4.2 more explicitly for $N=4$ . We have $m=6$ , $\varepsilon_{2}=\varepsilon_{3}=0$ . Let $r_{1}$ , $r_{2}$ , $r_{4}$ be rational numbers and set $k=\frac{1}{2}(r_{1}+r_{2}+r_{4})$ . Let

(44)

\begin{pmatrix}x_{1}\\ x_{2}\\ x_{4}\end{pmatrix}=\begin{pmatrix}4&2&1\\ 1&2&1\\ 1&2&4\end{pmatrix}\cdot\begin{pmatrix}r_{1}\\ r_{2}\\ r_{4}\end{pmatrix}.

Our aim is to find an Eisenstein series $E(\tau)$ of weight $k$ equal to $f(\tau)=\eta(\tau)^{r_{1}}\eta(2\tau)^{r_{2}}\eta(4\tau)^{r_{4}}$ . We only consider the Eisenstein series defined at the cusp $\mathrm{i}\infty$ , so assume $x_{4}=0$ which means $f$ does not vanish at the cusp $\mathrm{i}\infty$ . Since $f$ must be a holomorphic eta-quotient, we assume $x_{1},x_{2}\geq 0$ . Therefore,

x_{1}=-6r_{2}-15r_{4}\geq 0,\quad x_{2}=-3r_{4}\geq 0,\quad\text{hence }k=\frac{1}{2}(-r_{2}-3r_{4})\geq 0.

Now the first inequality in (21) becomes $k>-4+2\left(\left\{\frac{x_{1}}{24}\right\}+\left\{\frac{x_{2}}{24}\right\}\right)$ , which holds since $k\geq 0$ . Consequently, (22) with $t=0$ is applicable. We summarize:

Proposition 8.1.

Let $r_{2}$ and $r_{4}$ be rational numbers satisfying $-2r_{2}-5r_{4}\geq 0$ and $-r_{4}\geq 0$ . Let $D\in\mathbb{Z}_{\geq 1}$ such that $Dr_{2},\,Dr_{4}\in 2\mathbb{Z}$ . Then we have

\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(4),\chi)=\left[\frac{-2r_{2}-5r_{4}}{8}\right]+\left[\frac{-r_{4}}{8}\right]+1,

where $\chi\colon\widetilde{\Gamma_{0}(4)^{D}}\to\mathbb{C}^{\times}$ is the character of $\eta(\tau)^{-2r_{2}-4r_{4}}\eta(2\tau)^{r_{2}}\eta(4\tau)^{r_{4}}$ and $k=\frac{1}{2}(-r_{2}-3r_{4})$ .

Proof.

We have shown that (22) with $t=0$ holds. A simplification gives the desired formula. ∎

Corollary 8.2.

Let the notation and assumptions be as in Proposition 8.1. Then $\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(4),\chi)=1$ if and only if

0\leq-2r_{2}-5r_{4}<8\quad\text{and}\quad 0\leq-r_{4}<8.

Moreover, suppose $\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(4),\chi)=1$ ; then $k>2$ if and only if $(-2r_{2}-5r_{4})+(-r_{4})>8$ .

Note that when $\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(4),\chi)=1$ , then $M_{k}(\Gamma_{0}(4),\chi)=\mathbb{C}\eta(\tau)^{-2r_{2}-4r_{4}}\eta(2\tau)^{r_{2}}\eta(4\tau)^{r_{4}}$ . Hence if $M_{k}(\Gamma_{0}(4),\chi)$ contains an Eisenstein series, then an identity that relates the eta-quotient and the Eisenstein series follows. In the rest we only consider the case $k>2$ in which nonholomorphic Eisenstein series are avoided. Similar identities can also be derived in the cases $k<2$ , which we will do in the future work.

Recall the definition of Eisenstein series of rational weights greater than $2$ :

Definition 8.3 (Definition 3.1, [16]).

Let $D,N$ be positive integers, and let $k\in\frac{1}{D}\mathbb{Z}$ that is greater than $2$ . Let $\chi\colon\widetilde{\Gamma_{0}(N)^{D}}\to\mathbb{C}^{\times}$ be a finite index character such that $\chi(\widetilde{-I})=\mathfrak{e}\left(-k/2\right)$ . Let $s\in\mathbb{P}^{1}(\mathbb{Q})$ and $\gamma_{s}\in\mathrm{SL}_{2}(\mathbb{Z})$ such that $\gamma_{s}(\mathrm{i}\infty)=s$ . If $\chi(\widetilde{\gamma_{s}}\widetilde{T}^{w_{s}}\widetilde{\gamma_{s}}^{-1})=1$ , then we define the Eisenstein series $E_{\gamma_{s},k}$ on the group $\widetilde{\Gamma_{0}(N)^{D}}$ , of weight $k$ , with character $\chi$ and at cusp $\gamma_{s}$ as

E_{\gamma_{s},k}(\tau)=\sum_{\gamma\in\widetilde{\gamma_{s}}\langle\widetilde{T}^{w_{s}}\rangle\widetilde{\gamma_{s}}^{-1}\backslash\widetilde{\Gamma_{0}(N)^{D}}}\chi(\gamma)^{-1}\cdot 1|_{k}\widetilde{\gamma_{s}}^{-1}\gamma.

In the above definition, $w_{s}$ is the width (c.f. Section 3). If $s=a/c$ where $a,c$ are coprime integers, then $w_{s}=\frac{N}{(N,c^{2})}$ . The basic properties of these $E_{\gamma_{s},k}$ are collected in [16, Section 3].

Now let $N=4$ and let $\chi$ be as in Proposition 8.1. Since $\chi(\widetilde{T})=1$ the Eisenstein series $E_{I,k}(\tau)$ is well-defined provided that $k>2$ , that is, $(-2r_{2}-5r_{4})+(-r_{4})>8$ . The main result of this section is the following:

Theorem 8.4.

Let $r_{2}$ and $r_{4}$ be rational numbers satisfying

(45)

0\leq-2r_{2}-5r_{4}<8,\quad 0\leq-r_{4}<8,\quad(-2r_{2}-5r_{4})+(-r_{4})>8.

Let $D$ be a positive integer such that $Dr_{2},\,Dr_{4}\in 2\mathbb{Z}$ . Let $\chi\colon\widetilde{\Gamma_{0}(4)^{D}}\to\mathbb{C}^{\times}$ be the character of $\eta(\tau)^{-2r_{2}-4r_{4}}\eta(2\tau)^{r_{2}}\eta(4\tau)^{r_{4}}$ . Set $k=\frac{1}{2}(-r_{2}-3r_{4})$ and

E_{I,k}(\tau)=E_{I,k}(\tau;r_{2},r_{4})=\sum_{\gamma\in\langle\widetilde{T}\rangle\backslash\widetilde{\Gamma_{0}(4)^{D}}}\chi(\gamma)^{-1}\cdot 1|_{k}\gamma.

Then we have

(46)

\eta(\tau)^{-2r_{2}-4r_{4}}\eta(2\tau)^{r_{2}}\eta(4\tau)^{r_{4}}=\frac{1}{2D}E_{I,k}(\tau;r_{2},r_{4}).

Proof.

Corollary 8.2 shows that $\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(4),\chi)=1$ . Let $x_{1}$ , $x_{2}$ and $x_{4}$ be as in (44) with $r_{1}=-2r_{2}-4r_{4}$ . Then the assumptions imply that $x_{1}\geq 0$ , $x_{2}\geq 0$ and $x_{4}=0$ . Thus $\eta(\tau)^{-2r_{2}-4r_{4}}\eta(2\tau)^{r_{2}}\eta(4\tau)^{r_{4}}$ is holomorphic at all cusps and hence is in $M_{k}(\Gamma_{0}(4),\chi)$ . By [16, Theorem 3.3] we have $E_{I,k}\in M_{k}(\Gamma_{0}(4),\chi)$ . It follows that $\eta(\tau)^{-2r_{2}-4r_{4}}\eta(2\tau)^{r_{2}}\eta(4\tau)^{r_{4}}=c\cdot E_{I,k}$ . Let $\Im\tau\to+\infty$ ; we find that $c=\frac{1}{2D}$ again by [16, Theorem 3.3]. ∎

In the rest of this section we will write out the Fourier expansion of $\frac{1}{2D}E_{I,k}(\tau)$ which will give us a more explicit form of the identity (46).

Lemma 8.5.

For any rational numbers $r_{2},r_{4}$ with $k=\frac{1}{2}(-r_{2}-3r_{4})>2$ we have

\frac{1}{2D}E_{I,k}(\tau;r_{2},r_{4})=1+\sum_{\begin{subarray}{c}{4\mid c>0,d\in\mathbb{Z}}\\ {(c,d)=1}\end{subarray}}\mathfrak{e}\left(-\frac{1}{24}P(c,d;r_{2},r_{4})\right)\cdot(c\tau+d)^{-k}

where

	$\displaystyle P(c,d;r_{2},r_{4})$	$\displaystyle=r_{2}\cdot\left(\Psi\begin{pmatrix}{a}&{2b}\\ {c/2}&{d}\end{pmatrix}-2\Psi\begin{pmatrix}{a}&{b}\\ {c}&{d}\end{pmatrix}\right)+r_{4}\cdot\left(\Psi\begin{pmatrix}{a}&{4b}\\ {c/4}&{d}\end{pmatrix}-4\Psi\begin{pmatrix}{a}&{b}\\ {c}&{d}\end{pmatrix}\right)$
(47)			$\displaystyle=r_{2}\cdot\left(12s(-d,c/2)-24s(-d,c)+3\right)+r_{4}\cdot\left(12s(-d,c/4)-48s(-d,c)+9\right)$

with $a,b$ being any integers such that $ad-bc=1$ .

Proof.

According to [16, Lemma 3.2],

E_{I,k}(\tau;r_{2},r_{4})=D\cdot\sum_{\begin{subarray}{c}{4\mid c\in\mathbb{Z},d\in\mathbb{Z}}\\ {(c,d)=1}\end{subarray}}\chi\widetilde{\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right)}^{-1}\cdot(c\tau+d)^{-k}

where $a,b$ are any integers such that $ad-bc=1$ . By Remark 2.2, $\chi\left(\widetilde{-I}\right)=\mathfrak{e}\left(-k/2\right)$ . It follows that

\chi\widetilde{\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right)}^{-1}\cdot(c\tau+d)^{-k}=\chi\widetilde{\left(\begin{smallmatrix}{-a}&{-b}\\ {-c}&{-d}\end{smallmatrix}\right)}^{-1}\cdot(-c\tau-d)^{-k}.

Thus

\frac{1}{2D}E_{I,k}(\tau;r_{2},r_{4})=1+\sum_{\begin{subarray}{c}{4\mid c>0,d\in\mathbb{Z}}\\ {(c,d)=1}\end{subarray}}\chi\widetilde{\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right)}^{-1}\cdot(c\tau+d)^{-k}.

Substituting (6) and (5) in the above expression gives the desired formula. ∎

Proposition 8.6.

With the notation being as above, we have

\frac{1}{2D}E_{I,k}(\tau;r_{2},r_{4})=1+\mathrm{e}^{-\uppi\mathrm{i}k/2}\frac{(2\uppi)^{k}}{\Gamma(k)}\sum_{n\in\mathbb{Z}_{\geq 1}}n^{k-1}a(n)\cdot q^{n},

where

a(n)=\sum_{c\in\mathbb{Z}_{\geq 1}}\frac{1}{(4c)^{k}}\sum_{\begin{subarray}{c}{0\leq d<4c}\\ {(d,4c)=1}\end{subarray}}\mathfrak{e}\left(\frac{dn}{4c}-\frac{1}{24}P(4c,d;r_{2},r_{4})\right).

Proof.

For $d\in\mathbb{Z}$ , $c\in\mathbb{Z}_{\geq 1}$ set

f(d,c)=\begin{dcases}\mathfrak{e}\left(-\frac{1}{24}P(c,d;r_{2},r_{4})\right)&\text{ if }4\mid c\text{ and }(d,c)=1;\\ 0&\text{ if }4\nmid c\text{ and }(d,c)=1;\\ f\left(\frac{d}{(d,c)},\frac{c}{(d,c)}\right)&\text{ if }(d,c)>1.\end{dcases}

Thus, $\frac{1}{2D}E_{I,k}(\tau;r_{2},r_{4})=1+\zeta(k)^{-1}\cdot\sum_{c\in\mathbb{Z}_{\geq 1},d\in\mathbb{Z}}f(d,c)(c\tau+d)^{-k}$ since $\sum_{t\mid(c,d)}\mu(t)=0$ if $(c,d)>1$ and $\sum_{t}\frac{\mu(t)}{t^{k}}=\zeta(k)^{-1}$ , where $\zeta$ is the Riemann zeta function and $\mu$ is the Möbius function. Note that $f(d+c,c)=f(d,c)$ . This fact, together with the Lipschitz summation formula

\sum_{n\in\mathbb{Z}}\frac{1}{(\tau+n)^{s}}=\mathrm{e}^{-\uppi\mathrm{i}s/2}\frac{(2\uppi)^{s}}{\Gamma(s)}\sum_{n\geq 1}n^{s-1}\mathrm{e}^{2\uppi\mathrm{i}n\tau},\quad\tau\in\mathfrak{H},\,\Re(s)>1

implies that

	$\displaystyle\sum_{d\in\mathbb{Z}}f(d,c)(c\tau+d)^{-k}$	$\displaystyle=c^{-k}\sum_{0\leq d_{1}<c}f(d_{1},c)\sum_{d_{0}\in\mathbb{Z}}(\tau+\frac{d_{1}}{c}+d_{0})^{-k}$
		$\displaystyle=\mathrm{e}^{-\uppi\mathrm{i}k/2}\frac{(2\uppi)^{k}}{\Gamma(k)}c^{-k}\sum_{n\in\mathbb{Z}_{\geq 1}}n^{k-1}\sum_{0\leq d<c}f(d,c)\mathfrak{e}\left(\frac{dn}{c}\right)q^{n}.$

Therefore,

(48)

\frac{1}{2D}E_{I,k}(\tau;r_{2},r_{4})=1+\mathrm{e}^{-\uppi\mathrm{i}k/2}\frac{(2\uppi)^{k}}{\Gamma(k)\zeta(k)}\sum_{n\in\mathbb{Z}_{\geq 1}}n^{k-1}\sum_{c\in\mathbb{Z}_{\geq 1}}\frac{1}{c^{k}}\sum_{0\leq d<c}f(d,c)\mathfrak{e}\left(\frac{dn}{c}\right)q^{n}.

By the definition of $f(d,c)$ we find that

	$\displaystyle\sum_{c\in\mathbb{Z}_{\geq 1}}\frac{1}{c^{k}}\sum_{0\leq d<c}f(d,c)\mathfrak{e}\left(\frac{dn}{c}\right)$	$\displaystyle=\sum_{t\in\mathbb{Z}_{\geq 1}}\sum_{\begin{subarray}{c}{c\in\mathbb{Z}_{\geq 1},0\leq d<c}\\ {(d,c)=t}\end{subarray}}\frac{1}{(tc)^{k}}f(td,tc)\mathfrak{e}\left(\frac{dn}{c}\right)$
		$\displaystyle=\zeta(k)\cdot\sum_{\begin{subarray}{c}{c\in\mathbb{Z}_{\geq 1},0\leq d<4c}\\ {(d,4c)=1}\end{subarray}}\frac{1}{(4c)^{k}}f(d,4c)\mathfrak{e}\left(\frac{dn}{4c}\right)$
		$\displaystyle=\zeta(k)\cdot\sum_{\begin{subarray}{c}{c\in\mathbb{Z}_{\geq 1},0\leq d<4c}\\ {(d,4c)=1}\end{subarray}}\frac{1}{(4c)^{k}}\mathfrak{e}\left(\frac{dn}{4c}-\frac{1}{24}P(4c,d;r_{2},r_{4})\right).$

Inserting this into (48) gives the desired formula. ∎

Remark 8.7.

The reader may compare the above proposition with [16, Theorem 6.2].

We can now give an equivalent statement of Theorem 8.4 in a form that gives Fourier coefficients of certain infinite $q$ -products:

Corollary 8.8.

Let $r_{2}$ and $r_{4}$ be rational numbers satisfying (45). Let $D$ be a positive integer such that $Dr_{2},\,Dr_{4}\in 2\mathbb{Z}$ . Let $\sum_{n\in\mathbb{Z}_{\geq 0}}c(n)q^{n}$ be the holomorphic branch of the $D$ -th root of the infinite product

\prod_{n\in\mathbb{Z}_{\geq 1}}(1-q^{n})^{-2Dr_{2}-4Dr_{4}}\prod_{n\in\mathbb{Z}_{\geq 1}}(1-q^{2n})^{Dr_{2}}\prod_{n\in\mathbb{Z}_{\geq 1}}(1-q^{4n})^{Dr_{4}}

with $c(0)=1$ . Then for $n\geq 1$ we have

c(n)=\mathrm{e}^{-\uppi\mathrm{i}k/2}\frac{(2\uppi)^{k}}{\Gamma(k)}n^{k-1}\sum_{c\in\mathbb{Z}_{\geq 1}}\frac{1}{(4c)^{k}}\sum_{\begin{subarray}{c}{0\leq d<4c}\\ {(d,4c)=1}\end{subarray}}\mathfrak{e}\left(\frac{dn}{4c}-\frac{1}{24}P(4c,d;r_{2},r_{4})\right).

(See (47) for the definition of $P(4c,d;r_{2},r_{4})$ .)

Proof.

There are exactly $D$ holomorphic branches and the one with the leading term $1$ is the eta-quotient $\eta(\tau)^{-2r_{2}-4r_{4}}\eta(2\tau)^{r_{2}}\eta(4\tau)^{r_{4}}$ . Now the assertion follows from Theorem 8.4 and Proposition 8.6. ∎

Example 8.9.

Let us consider the special case $r_{2}=0$ , $r_{4}=-\frac{3}{2}$ for which (45) holds. Note that $k=\frac{1}{2}(-r_{2}-3r_{4})=\frac{9}{4}$ . Set $D=4$ . Then Corollary 8.8 shows

\frac{\prod_{n\in\mathbb{Z}_{\geq 1}}(1-q^{n})^{6}}{\sqrt{\prod_{n\in\mathbb{Z}_{\geq 1}}(1-q^{4n})^{3}}}=1+\mathrm{e}^{-9\uppi\mathrm{i}/8}\frac{(2\uppi)^{\frac{9}{4}}}{\Gamma(\frac{9}{4})}\sum_{n\in\mathbb{Z}_{\geq 1}}n^{\frac{5}{4}}\left(\sum_{c\in\mathbb{Z}_{\geq 1}}\frac{1}{(4c)^{\frac{9}{4}}}\sum_{\begin{subarray}{c}{0\leq d<4c}\\ {(d,4c)=1}\end{subarray}}\mathfrak{e}\left(\frac{dn}{4c}-\frac{1}{24}P(4c,d;0,-\frac{3}{2})\right)\right)\cdot q^{n},

where the square root is the principal branch $\sqrt{\prod_{n\in\mathbb{Z}_{\geq 1}}(1-q^{4n})^{3}}=1+o(1)$ and

P(4c,d;0,-\frac{3}{2})=-18s(-d,c)+72s(-d,4c)-\frac{27}{2}.

(See (4) for the definition of $s(-d,c)$ .) For numerical check (of this and the next examples) using SageMath programs, see Appendix A.

Example 8.10.

Corollary 8.8 contains as well some identities involving ordinary eta-quotients, that is, eta-quotients with integral exponents. For instance, set $r_{2}=7$ , $r_{4}=-4$ and hence $k=\frac{1}{2}(-r_{2}-3r_{4})=\frac{5}{2}$ and $D=2$ . For this setting, (45) holds. The identity reads

\prod_{n\in\mathbb{Z}_{\geq 1}}(1-q^{n})^{2}(1-q^{2n})^{7}(1-q^{4n})^{-4}\\ =1+\mathrm{e}^{-5\uppi\mathrm{i}/4}\frac{(2\uppi)^{\frac{5}{2}}}{\Gamma(\frac{5}{2})}\sum_{n\in\mathbb{Z}_{\geq 1}}n^{\frac{3}{2}}\sum_{c\in\mathbb{Z}_{\geq 1}}\frac{1}{(4c)^{\frac{5}{2}}}\sum_{\begin{subarray}{c}{0\leq d<4c}\\ {(d,4c)=1}\end{subarray}}\mathfrak{e}\left(\frac{dn}{4c}-\frac{1}{24}P(4c,d;7,-4)\right)\cdot q^{n}

where

P(4c,d;7,-4)=-48s(-d,c)+84s(-d,2c)+24s(-d,4c)-15.

We will return to this function, i.e., $\eta(\tau)^{2}\eta(2\tau)^{7}\eta(4\tau)^{-4}$ in the next section (c.f. Section 9.4.1) and show that it is, among others, a Hecke eigenform in a generalized sense. (In [20, Section 4], the authors obtained some Hecke eigenforms in the generalized sense. However, for $N=4$ , only Hecke eigenforms of weight $1/2$ , $1$ , $3/2$ are obtained. The current example gives a Hecke eigenform of weight $5/2$ which can not be achieved using methods in [20].)

9. Application: an extension of Martin’s list of multiplicative eta-quotients

In 1996 Martin [17] obtained the complete list of integral-weight holomorphic eta-quotients that are eigenforms for all Hecke operators. This generalized a previous result of Dummit, Kisilevsky and McKay [41]. The term “eta-quotient” in [17] refers to one whose multiplier system is induced by a Dirichlet character (necessarily real) and the Hecke operators are in the sense of [18].

In this section, we provide an algorithm for finding out all holomorphic eta-quotients $f(\tau)=\prod_{n\geq 1}\eta(n\tau)^{r_{n}}$ of integral exponents $r_{n}$ (integral or half-integral weights) and of arbitrary multiplier systems such that either of the following conditions holds:

(a)

$\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(N),\chi)$ can be computed using Theorem 4.2 (with $t=0$ ) and is equal to $1$ ,
(b)

$\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(N),\chi)$ can be computed using Theorem 4.2 (with $t=0$ ) and is greater than $1$ , $f$ is a cusp form, $k>2$ and there exist $\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(N),\chi)-1$ linearly independent Eisenstein series in $M_{k}(\Gamma_{0}(N),\chi)$ (c.f. Definition 8.3),

where $k=\frac{1}{2}\sum_{n}r_{n}$ is the weight, $N$ is the least common multiple of all $n$ with $r_{n}\neq 0$ (when $r_{n}=0$ for all $n$ , let $N=1$ ) and $\chi$ is the character $\widetilde{\Gamma_{0}(N)}\rightarrow\mathbb{C}^{\times}$ of $f$ . (Note that throughout this section $\widetilde{\Gamma_{0}(N)}=\widetilde{\Gamma_{0}(N)^{2}}$ as in Section 6.)

If $f$ satisfies either of the above conditions, we say it is admissible. If $f$ satisfies the condition (a) then we say it is admissible of type I; if it satisfies (b) then we say it is admissible of type II. Moreover, if $f$ satisfies either of the conditions

(a’)

$\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(N),\chi)=1$ ,
(b’)

$\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(N),\chi)\geq 2$ but $\dim_{\mathbb{C}}S_{k}(\Gamma_{0}(N),\chi)=1$ and $f$ is a cusp form,

we say $f$ is weakly admissible. Thus an admissible eta-quotient must be a weakly admissible eta-quotient. Similarly, we can define the concept of weakly admissible eta-quotients of type I and type II. For each weakly admissible $f$ , we will show that it is a Hecke eigenform in the sense of Wohlfahrt [19]; c.f. Theorem 9.19. Therefore, our list of admissible eta-quotients can be considered as an extension of Martin’s list.

9.1. The algorithm for finding admissible eta-quotients

Throughout this subsection, let $f(\tau)=\prod_{n\geq 1}\eta(n\tau)^{r_{n}}$ be a holomorphic eta-quotient where $r_{n}$ are integers and are equal to $0$ for all but finitely many $n$ . Let $k=\frac{1}{2}\sum_{n}r_{n}$ be the weight and $N$ , the level⁹⁹9The reader should notice that this usage of “eta quotient of level $N$ ” is different from that in the second half of Section 2. In the usage here any eta-quotient has a uniquely determined level while in the usage before, an eta-quotient, say $\eta(\tau)\eta(2\tau)$ , may be regarded as of level $2$ , or of level $4$ , etc., be the least common multiple of $\{n\in\mathbb{Z}_{\geq 1}\colon r_{n}\neq 0\}$ (with the convention $N=1$ if $r_{n}=0$ for all $n$ ). Let $\chi\colon\widetilde{\Gamma_{0}(N)}\rightarrow\mathbb{C}^{\times}$ be the character of $f$ (c.f. (6) with $D=2$ ). Since $f$ is holomorphic at all cusps we have $x_{c}\geq 0$ for all $c\mid N$ ( $x_{c}$ are defined in (19)). By Remark 4.3 we have

	$\displaystyle\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(N),\chi)$	$\displaystyle\geq\frac{k-1}{12}m+\frac{1}{4}\varepsilon_{2}+\frac{1}{3}\varepsilon_{3}+\sum_{c\mid N}\phi(c,N/c)\cdot\left(\frac{1}{2}-\left\{\frac{x_{c}}{24}\right\}\right)$
		$\displaystyle>\frac{k-1}{12}m-\frac{1}{2}\sum_{c\mid N}\phi(c,N/c)$

where $m$ , $\varepsilon_{2}$ , $\varepsilon_{3}$ are given by (9), (10), (11) respectively.

Lemma 9.1.

If $\left(\frac{(k-1)N}{12}-\frac{3\sqrt{N}}{2}\right)\cdot\prod_{p\mid N}\left(1+\frac{1}{p}\right)\geq 1$ , then $f$ is not admissible. As a consequence, there are only finitely many admissible eta-quotients.

Proof.

Assume by contradiction that $f$ is admissible. If the condition (a) holds, then

(49)

1=\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(N),\chi)>\frac{k-1}{12}m-\frac{1}{2}\sum_{c\mid N}\phi(c,N/c).

By (41) we have $\varepsilon_{\infty}=\sum_{c\mid N}\phi(c,N/c)\leq m\cdot N^{-1/2}$ . This, together with (49) and (9), implies that $\left(\frac{(k-1)N}{12}-\frac{\sqrt{N}}{2}\right)\cdot\prod_{p\mid N}\left(1+\frac{1}{p}\right)<1$ which contradicts the assumption. On the other hand, if the condition (b) holds. Since $k>2$ , there are totally¹⁰¹⁰10These $n_{0}$ functions, in the rational weight ( $k>2$ ) situation, constitute a basis of the Eisenstein space which, as in the integral weight case, is defined as the orthogonal complement of the space of cusp forms under the Petersson inner product. In addition, one can prove that $n_{0}=\lvert R\rvert$ where $R$ is (18) with $f_{0}$ arbitrarily chosen. $n_{0}$ Eisenstein series in $M_{k}(\Gamma_{0}(N),\chi)$ , where

(50)

n_{0}=\mathop{\mathrm{\#}}\left\{s\in\Gamma_{0}(N)\backslash\mathbb{P}^{1}(\mathbb{Q})\colon\chi(\widetilde{\gamma_{s}}\widetilde{T}^{w_{s}}\widetilde{\gamma_{s}}^{-1})=1\right\},

( $w_{s}$ is the width; c.f. Section 3, $\gamma_{s}$ is any matrix in $\mathrm{SL}_{2}(\mathbb{Z})$ such that $\gamma_{s}(\mathrm{i}\infty)=s$ ) and these $n_{0}$ functions are $\mathbb{C}$ -linearly independent. See [16, Theorem 3.3]. By condition (b) we have $n_{0}\geq\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(N),\chi)-1$ . Since in the current case $f$ is a cusp form, these $n_{0}$ Eisenstein series and $f$ together constitute a basis of $M_{k}(\Gamma_{0}(N),\chi)$ and $n_{0}=\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(N),\chi)-1$ . Hence

(51)

n_{0}+1=\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(N),\chi)>\frac{k-1}{12}m-\frac{1}{2}\sum_{c\mid N}\phi(c,N/c).

This inequality, together with (41), (9) and the fact $n_{0}\leq\varepsilon_{\infty}=\sum_{c\mid N}\phi(c,N/c)$ , implies that $\left(\frac{(k-1)N}{12}-\frac{3\sqrt{N}}{2}\right)\cdot\prod_{p\mid N}\left(1+\frac{1}{p}\right)<1$ which contradicts the assumption. Therefore, we have shown that both of conditions (a) and (b) lead to contradictions, so $f$ is not admissible.

Now we begin to prove that there are only finitely many admissible eta-quotients. Since $f\in M_{k}(\Gamma_{0}(N),\chi)$ is a holomorphic eta-quotient of integral exponents, $2k\in\mathbb{Z}_{\geq 0}$ . For $k=0$ , we must have $f=1$ . For $k=1/2$ and $1$ , we have shown in Remark 7.2 that there are only finitely many $N$ such that $\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(N),\chi)$ can be calculated using Theorem 4.2. For each pair $(k,N)$ , the holomorphic eta-quotients of level $N$ and weight $k$ constitute a finite set since the cardinality of this set does not exceed the cardinality of the solution set of $\sum_{c\mid N}\phi(c,N/c)x_{c}=2mk$ , $x_{c}\in\mathbb{Z}_{\geq 0}$ (c.f. Proposition 5.1 and Remark 5.2) which is finite. This proves that there are only finitely many admissible eta-quotients of weights $0$ , $1/2$ and $1$ . For each weight $k\geq 3/2$ , there exists an $N_{0}$ such that when $N>N_{0}$ we have $\left(\frac{(k-1)N}{12}-\frac{3\sqrt{N}}{2}\right)\cdot\prod_{p\mid N}\left(1+\frac{1}{p}\right)\geq 1$ . Thus, the level of an admissible eta-quotient of weight $k$ does not exceed $N_{0}$ and hence there are finitely many of them. Finally, if $k\geq 19$ and $N\geq 2$ , we have

\left(\frac{(k-1)N}{12}-\frac{3\sqrt{N}}{2}\right)\cdot\prod_{p\mid N}\left(1+\frac{1}{p}\right)\geq\frac{3}{2}(N-\sqrt{N})\cdot\prod_{p\mid N}\left(1+\frac{1}{p}\right)\geq 1,

and if $k\geq 31$ , $N=1$ , then the left-hand side still $\geq 1$ . It follows that if the weight of $f$ is greater than or equal to $31$ , then $f$ is not admissible. Therefore, there are only finitely many admissible eta-quotients. ∎

Corollary 9.2.

If $N>400$ , then $f$ is not admissible. Moreover, if $N>36$ , then $f$ is not admissible of type I.

Proof.

Suppose $N>400$ ; we have $k\neq 0$ . If $k=1/2$ , $1$ or $3/2$ , then by Tables 1, LABEL:table:wt1 and Section 7.3, $\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(N),\chi)$ can not be calculated using Theorem 4.2 with $t=0$ (that is, the first inequality in (21) does not hold) for any $\chi$ . Thus, $f$ is not admissible¹¹¹¹11Although $f$ is possibly weakly admissible.. Otherwise, if $k\geq 2$ , then obviously $\left(\frac{(k-1)N}{12}-\frac{3\sqrt{N}}{2}\right)\cdot\prod_{p\mid N}\left(1+\frac{1}{p}\right)\geq 1$ . It follows from this and Lemma 9.1 that $f$ is not admissible.

Now we begin to prove the latter assertion; suppose $N>36$ . Certainly $k\neq 0$ . According to Table 1, Table LABEL:table:wt1 and Subsection 7.3, if $k=1/2$ , $1$ or $3/2$ , then $\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(N),\chi)\geq 2$ provided it can be calculated using Theorem 4.2 (with $t=0$ ). Hence $f$ is not admissible of type I. If $k\geq 2$ , then for $36<N\leq 400$ but $N\neq 49$ we have

\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(N),\chi)>\frac{k-1}{12}m-\frac{1}{2}\sum_{c\mid N}\phi(c,N/c)>1.

See Appendix A for the SageMath code verifying this. If $N=49$ , then $\varepsilon_{3}=2$ (c.f. (11)), and hence

	$\displaystyle\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(49),\chi)$	$\displaystyle\geq\frac{k-1}{12}m+\frac{1}{4}\varepsilon_{2}+\frac{1}{3}\varepsilon_{3}+\sum_{c\mid N}\phi(c,N/c)\cdot\left(\frac{1}{2}-\left\{\frac{x_{c}}{24}\right\}\right)$
		$\displaystyle>\frac{2-1}{12}\cdot 56+\frac{2}{3}-\frac{1}{2}\sum_{c\mid 49}\phi(c,49/c)>1.$

Therefore $f$ is not admissible of type I for $k\geq 2$ , $36<N\leq 400$ . ∎

According to the above corollary and Lemma 9.1, the following algorithm can be used to present all admissible nonconstant eta-quotients.

(a)

Let $N$ range over $\{1,2,3,4,\dots,400\}$ .
(b)

For a fixed $N$ , set $K_{N}=\left\{k\in\frac{1}{2}\mathbb{Z}\colon\frac{1}{2}\leq k<1+\frac{12}{m}+\frac{18}{\sqrt{N}}\right\}$ , where $m$ is defined in (9).
(c)

Let $N$ be fixed as above. For any $c\mid N$ , let $x_{c}$ range over nonnegative integers such that $\sum_{c\mid N}\phi(c,N/c)x_{c}\in 2m\cdot K_{N}$ .
(d)

Let $N$ and $(x_{c})_{c\mid N}$ be fixed. Determine the sequence $(r_{n})_{n\mid N}$ by (27) and set $k=\frac{1}{2}\sum_{n\mid N}r_{n}$ , $g=\prod_{n\mid N}\eta(n\tau)^{r_{n}}$ . Since $\sum_{c\mid N}\phi(c,N/c)x_{c}=m\cdot\sum_{n\mid N}r_{n}$ (c.f. Remark 5.2) we have $k\in K_{N}$ .
(e)

If $r_{n}$ are not all integers, or they are all integers but $N$ is not the least common multiple of $\{r_{n}\neq 0\colon n\mid N\}$ , then we continue to the next value of $(x_{c})_{c\mid N}$ in Step (d) since in these cases either $g$ is not an eta-quotient of integral exponents or $g$ has been considered in a previous step.
(f)

We check whether (21) with $t=0$ holds. If it does not hold, $g$ is not admissible so we go to Step (d) and consider the next value of $(x_{c})_{c\mid N}$ .
(g)

We check whether $\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(N),\chi)=1$ via (22). If this is the case, then we find an admissible $g$ . We record it, go to Step (d) and consider the next value of $(x_{c})_{c\mid N}$ .
(h)

We check whether $k>2$ , $g$ is a cusp form (equivalent to $x_{c}>0$ for all $c\mid N$ ) and $\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(N),\chi)=n_{0}+1$ where $n_{0}$ is given in (50). If this is the case, then $g$ is admissible which we record; otherwise $g$ is not admissible. We have completely determined whether this $g$ is admissible or not so we go to Step (d) and consider the next value of $(x_{c})_{c\mid N}$ .
(i)

Since for each $N$ there are only finitely many values of $(x_{c})_{c\mid N}$ , this algorithm halts in finitely many steps.

Proposition 9.3.

The above algorithm gives all admissible nonconstant eta-quotients with no repetition.

Proof.

First, it is immediate that the functions $g$ recorded in Steps (g) or (h) are admissible nonconstant eta-quotients. Second, suppose $g_{1}$ and $g_{2}$ are functions recorded in Steps (g) or (h) corresponding to different pairs $\left(N_{1},(x_{c})_{c\mid N_{1}}\right)$ and $\left(N_{2},(x^{\prime}_{c})_{c\mid N_{2}}\right)$ . If $N_{1}\neq N_{2}$ , then the exponents of $g_{1}$ are different from those of $g_{2}$ since their least common multiples are different. Hence $g_{1}\neq g_{2}$ . Otherwise $N_{1}=N_{2}$ but their is some $c\mid N_{1}$ such that $x_{c}\neq x^{\prime}_{c}$ . Thus the sequences of exponents of $g_{1}$ and $g_{2}$ are different by Proposition 5.1, and hence $g_{1}\neq g_{2}$ as well. Finally, let $f=\prod_{n\mid N_{0}}\eta(n\tau)^{r_{n}}$ be any admissible nonconstant eta-quotient whose level is $N_{0}$ and weight is $k_{0}$ . By Corollary 9.2 we have $1\leq N_{0}\leq 400$ ; by Lemma 9.1 we have $k_{0}\in K_{N_{0}}$ . Define $(y_{c})_{c\mid N_{0}}$ by (19) (with $N=N_{0}$ and $x_{c}$ replaced by $y_{c}$ ). It is immediate that $y_{c}\in\mathbb{Z}$ . Moreover, $y_{c}\geq 0$ since $f$ is holomorphic at all cusps. We have $\sum_{c\mid N_{0}}\phi(c,N_{0}/c)y_{c}=2mk_{0}$ (c.f. Remark 5.2) and hence $\sum_{c\mid N_{0}}\phi(c,N_{0}/c)y_{c}\in 2m\cdot K_{N_{0}}$ where $m=N_{0}\prod_{p\mid N_{0}}(1+1/p)^{-1}$ . It follows that in Steps (a) and (c) the variable $N$ can actually take the value $N_{0}$ and $(x_{c})_{c\mid N}$ can take the value $(y_{c})_{c\mid N_{0}}$ . The corresponding $g$ in Step (d) is equal to $f$ according to Proposition 5.1. Therefore, $f$ is recorded in some step of the algorithm which concludes the proof. ∎

Remark 9.4.

If Step (h) is removed, then the remaining part of the algorithm gives all admissible nonconstant eta-quotients of type I. If this is the task, then we can shrink the set that $N$ ranges over in Step (a) to $\{1,2,3,\dots,36\}$ by the latter assertion of Corollary 9.2. In the following subsections, we will only consider such eta-quotients because finding those of type II need more effort and the program takes rather long time¹²¹²12We have provided the SageMath code that can produce admissible eta-quotients of type II. See Appendix A for the usage. which we will do in a future paper. The simplest example of type II is $\eta(\tau)^{24}$ since it is a cusp form, $\dim_{\mathbb{C}}M_{12}(\mathrm{SL}_{2}(\mathbb{Z}))=2$ and there is an Eisenstein series in $M_{12}(\mathrm{SL}_{2}(\mathbb{Z}))$ . As a comparison, $\eta(\tau)^{r}$ is an admissible eta-quotient of type I for $r=0,1,2,\dots,23$ since $\dim_{\mathbb{C}}M_{r/2}(\mathrm{SL}_{2}(\mathbb{Z}),\chi_{\eta^{r}})=1$ . See [20, Section 5].

9.2. The table of admissible eta-quotients of type I

For the SageMath program realizing the algorithm for finding admissible eta-quotients of type I in the last subsection, see Appendix A. Here (Table LABEL:table:admissibleTypeI below) we present all such eta-quotients for levels $N=1,\,p_{1},\,p_{1}^{2},\,p_{1}^{3}$ and $p_{1}p_{2}$ where $p_{1}$ , $p_{2}$ are distinct primes.

Table 3. All admissible eta-quotients of type I of level

N=1,p_{1},\,p_{1}^{2},\,p_{1}^{3}

p_{1}p_{2}

where

p_{1}

p_{2}

are distinct primes. An entry

n_{1}^{r_{1}}n_{2}^{r_{2}}\cdots n_{t}^{r_{t}}

represents the eta-quotient

\eta(n_{1}\tau)^{r_{1}}\eta(n_{2}\tau)^{r_{2}}\dots\eta(n_{t}\tau)^{r_{t}}

and

k

is the weight.

$N=1$
$1^{r}$ where $r=0,1,2,\dots,23$
$N=2$ , $k=1/2$
$1^{-1}2^{2}$	$1^{0}2^{1}$	$1^{2}2^{-1}$
$N=2$ , $k=1$
$1^{-2}2^{4}$	$1^{-1}2^{3}$	$1^{0}2^{2}$	$1^{1}2^{1}$	$1^{3}2^{-1}$	$1^{4}2^{-2}$
$N=2$ , $k=3/2$
$1^{-3}2^{6}$	$1^{-2}2^{5}$	$1^{-1}2^{4}$	$1^{0}2^{3}$	$1^{1}2^{2}$	$1^{2}2^{1}$
$1^{4}2^{-1}$	$1^{5}2^{-2}$	$1^{6}2^{-3}$
$N=2$ , $k=2$
$1^{-4}2^{8}$	$1^{-3}2^{7}$	$1^{-2}2^{6}$	$1^{-1}2^{5}$	$1^{0}2^{4}$	$1^{1}2^{3}$
$1^{2}2^{2}$	$1^{3}2^{1}$	$1^{5}2^{-1}$	$1^{6}2^{-2}$	$1^{7}2^{-3}$	$1^{8}2^{-4}$
$N=2$ , $k=5/2$
$1^{-5}2^{10}$	$1^{-4}2^{9}$	$1^{-3}2^{8}$	$1^{-2}2^{7}$	$1^{-1}2^{6}$	$1^{0}2^{5}$
$1^{1}2^{4}$	$1^{2}2^{3}$	$1^{3}2^{2}$	$1^{4}2^{1}$	$1^{6}2^{-1}$	$1^{7}2^{-2}$
$1^{8}2^{-3}$	$1^{9}2^{-4}$	$1^{10}2^{-5}$
$N=2$ , $k=3$
$1^{-6}2^{12}$	$1^{-5}2^{11}$	$1^{-4}2^{10}$	$1^{-3}2^{9}$	$1^{-2}2^{8}$	$1^{-1}2^{7}$
$1^{0}2^{6}$	$1^{1}2^{5}$	$1^{2}2^{4}$	$1^{3}2^{3}$	$1^{4}2^{2}$	$1^{5}2^{1}$
$1^{7}2^{-1}$	$1^{8}2^{-2}$	$1^{9}2^{-3}$	$1^{10}2^{-4}$	$1^{11}2^{-5}$	$1^{12}2^{-6}$
$N=2$ , $k=7/2$
$1^{-7}2^{14}$	$1^{-6}2^{13}$	$1^{-5}2^{12}$	$1^{-4}2^{11}$	$1^{-3}2^{10}$	$1^{-2}2^{9}$
$1^{-1}2^{8}$	$1^{0}2^{7}$	$1^{1}2^{6}$	$1^{2}2^{5}$	$1^{3}2^{4}$	$1^{4}2^{3}$
$1^{5}2^{2}$	$1^{6}2^{1}$	$1^{8}2^{-1}$	$1^{9}2^{-2}$	$1^{10}2^{-3}$	$1^{11}2^{-4}$
$1^{12}2^{-5}$	$1^{13}2^{-6}$	$1^{14}2^{-7}$
$N=2$ , $k=4$
$1^{-7}2^{15}$	$1^{-6}2^{14}$	$1^{-5}2^{13}$	$1^{-4}2^{12}$	$1^{-3}2^{11}$	$1^{-2}2^{10}$
$1^{-1}2^{9}$	$1^{0}2^{8}$	$1^{1}2^{7}$	$1^{2}2^{6}$	$1^{3}2^{5}$	$1^{4}2^{4}$
$1^{5}2^{3}$	$1^{6}2^{2}$	$1^{7}2^{1}$	$1^{9}2^{-1}$	$1^{10}2^{-2}$	$1^{11}2^{-3}$
$1^{12}2^{-4}$	$1^{13}2^{-5}$	$1^{14}2^{-6}$	$1^{15}2^{-7}$
$N=2$ , $k=9/2$
$1^{-5}2^{14}$	$1^{-4}2^{13}$	$1^{-3}2^{12}$	$1^{-2}2^{11}$	$1^{-1}2^{10}$	$1^{0}2^{9}$
$1^{1}2^{8}$	$1^{2}2^{7}$	$1^{3}2^{6}$	$1^{4}2^{5}$	$1^{5}2^{4}$	$1^{6}2^{3}$
$1^{7}2^{2}$	$1^{8}2^{1}$	$1^{10}2^{-1}$	$1^{11}2^{-2}$	$1^{12}2^{-3}$	$1^{13}2^{-4}$
$1^{14}2^{-5}$
$N=2$ , $k=5$
$1^{-3}2^{13}$	$1^{-2}2^{12}$	$1^{-1}2^{11}$	$1^{0}2^{10}$	$1^{1}2^{9}$	$1^{2}2^{8}$
$1^{3}2^{7}$	$1^{4}2^{6}$	$1^{5}2^{5}$	$1^{6}2^{4}$	$1^{7}2^{3}$	$1^{8}2^{2}$
$1^{9}2^{1}$	$1^{11}2^{-1}$	$1^{12}2^{-2}$	$1^{13}2^{-3}$
$N=2$ , $k=11/2$
$1^{-1}2^{12}$	$1^{0}2^{11}$	$1^{1}2^{10}$	$1^{2}2^{9}$	$1^{3}2^{8}$	$1^{4}2^{7}$
$1^{5}2^{6}$	$1^{6}2^{5}$	$1^{7}2^{4}$	$1^{8}2^{3}$	$1^{9}2^{2}$	$1^{10}2^{1}$
$1^{12}2^{-1}$
$N=2$ , $k=6$
$1^{1}2^{11}$	$1^{2}2^{10}$	$1^{3}2^{9}$	$1^{4}2^{8}$	$1^{5}2^{7}$	$1^{6}2^{6}$
$1^{7}2^{5}$	$1^{8}2^{4}$	$1^{9}2^{3}$	$1^{10}2^{2}$	$1^{11}2^{1}$
$N=2$ , $k=13/2$
$1^{3}2^{10}$	$1^{4}2^{9}$	$1^{5}2^{8}$	$1^{6}2^{7}$	$1^{7}2^{6}$	$1^{8}2^{5}$
$1^{9}2^{4}$	$1^{10}2^{3}$
$N=2$ , $k=7$
$1^{5}2^{9}$	$1^{6}2^{8}$	$1^{7}2^{7}$	$1^{8}2^{6}$	$1^{9}2^{5}$
$N=2$ , $k=15/2$
$1^{7}2^{8}$	$1^{8}2^{7}$
$N=3$ , $k=1/2$
$1^{0}3^{1}$
$N=3$ , $k=1$
$1^{-1}3^{3}$	$1^{0}3^{2}$	$1^{1}3^{1}$	$1^{3}3^{-1}$
$N=3$ , $k=3/2$
$1^{-1}3^{4}$	$1^{0}3^{3}$	$1^{1}3^{2}$	$1^{2}3^{1}$	$1^{4}3^{-1}$
$N=3$ , $k=2$
$1^{-2}3^{6}$	$1^{-1}3^{5}$	$1^{0}3^{4}$	$1^{1}3^{3}$	$1^{2}3^{2}$	$1^{3}3^{1}$
$1^{5}3^{-1}$	$1^{6}3^{-2}$
$N=3$ , $k=5/2$
$1^{-2}3^{7}$	$1^{-1}3^{6}$	$1^{0}3^{5}$	$1^{1}3^{4}$	$1^{2}3^{3}$	$1^{3}3^{2}$
$1^{4}3^{1}$	$1^{6}3^{-1}$	$1^{7}3^{-2}$
$N=3$ , $k=3$
$1^{-2}3^{8}$	$1^{-1}3^{7}$	$1^{0}3^{6}$	$1^{1}3^{5}$	$1^{2}3^{4}$	$1^{3}3^{3}$
$1^{4}3^{2}$	$1^{5}3^{1}$	$1^{7}3^{-1}$	$1^{8}3^{-2}$
$N=3$ , $k=7/2$
$1^{-1}3^{8}$	$1^{0}3^{7}$	$1^{1}3^{6}$	$1^{2}3^{5}$	$1^{3}3^{4}$	$1^{4}3^{3}$
$1^{5}3^{2}$	$1^{6}3^{1}$	$1^{8}3^{-1}$
$N=3$ , $k=4$
$1^{1}3^{7}$	$1^{2}3^{6}$	$1^{3}3^{5}$	$1^{4}3^{4}$	$1^{5}3^{3}$	$1^{6}3^{2}$
$1^{7}3^{1}$
$N=3$ , $k=9/2$
$1^{2}3^{7}$	$1^{3}3^{6}$	$1^{4}3^{5}$	$1^{5}3^{4}$	$1^{6}3^{3}$	$1^{7}3^{2}$
$N=3$ , $k=5$
$1^{4}3^{6}$	$1^{5}3^{5}$	$1^{6}3^{4}$
$N=3$ , $k=11/2$
$1^{5}3^{6}$	$1^{6}3^{5}$
$N=4$ , $k=1/2$
$1^{-2}2^{5}4^{-2}$	$1^{-1}2^{3}4^{-1}$	$1^{0}2^{-1}4^{2}$	$1^{0}2^{0}4^{1}$	$1^{0}2^{2}4^{-1}$	$1^{1}2^{-1}4^{1}$
$N=4$ , $k=1$
$1^{-4}2^{10}4^{-4}$	$1^{-3}2^{7}4^{-2}$	$1^{-3}2^{8}4^{-3}$	$1^{-2}2^{5}4^{-1}$	$1^{-2}2^{6}4^{-2}$	$1^{-2}2^{7}4^{-3}$
$1^{-1}2^{1}4^{2}$	$1^{-1}2^{2}4^{1}$	$1^{-1}2^{4}4^{-1}$	$1^{-1}2^{5}4^{-2}$	$1^{0}2^{-2}4^{4}$	$1^{0}2^{-1}4^{3}$
$1^{0}2^{0}4^{2}$	$1^{0}2^{1}4^{1}$	$1^{0}2^{3}4^{-1}$	$1^{0}2^{4}4^{-2}$	$1^{1}2^{-2}4^{3}$	$1^{1}2^{-1}4^{2}$
$1^{1}2^{0}4^{1}$	$1^{1}2^{2}4^{-1}$	$1^{2}2^{-2}4^{2}$	$1^{2}2^{-1}4^{1}$	$1^{2}2^{1}4^{-1}$	$1^{3}2^{-2}4^{1}$
$N=4$ , $k=3/2$
$1^{-6}2^{15}4^{-6}$	$1^{-5}2^{12}4^{-4}$	$1^{-5}2^{13}4^{-5}$	$1^{-4}2^{9}4^{-2}$	$1^{-4}2^{10}4^{-3}$	$1^{-4}2^{11}4^{-4}$
$1^{-4}2^{12}4^{-5}$	$1^{-3}2^{7}4^{-1}$	$1^{-3}2^{8}4^{-2}$	$1^{-3}2^{9}4^{-3}$	$1^{-3}2^{10}4^{-4}$	$1^{-2}2^{3}4^{2}$
$1^{-2}2^{4}4^{1}$	$1^{-2}2^{6}4^{-1}$	$1^{-2}2^{7}4^{-2}$	$1^{-2}2^{8}4^{-3}$	$1^{-2}2^{9}4^{-4}$	$1^{-1}2^{0}4^{4}$
$1^{-1}2^{1}4^{3}$	$1^{-1}2^{2}4^{2}$	$1^{-1}2^{3}4^{1}$	$1^{-1}2^{5}4^{-1}$	$1^{-1}2^{6}4^{-2}$	$1^{-1}2^{7}4^{-3}$
$1^{0}2^{-3}4^{6}$	$1^{0}2^{-2}4^{5}$	$1^{0}2^{-1}4^{4}$	$1^{0}2^{0}4^{3}$	$1^{0}2^{1}4^{2}$	$1^{0}2^{2}4^{1}$
$1^{0}2^{4}4^{-1}$	$1^{0}2^{5}4^{-2}$	$1^{0}2^{6}4^{-3}$	$1^{1}2^{-3}4^{5}$	$1^{1}2^{-2}4^{4}$	$1^{1}2^{-1}4^{3}$
$1^{1}2^{0}4^{2}$	$1^{1}2^{1}4^{1}$	$1^{1}2^{3}4^{-1}$	$1^{1}2^{4}4^{-2}$	$1^{2}2^{-3}4^{4}$	$1^{2}2^{-2}4^{3}$
$1^{2}2^{-1}4^{2}$	$1^{2}2^{0}4^{1}$	$1^{2}2^{2}4^{-1}$	$1^{2}2^{3}4^{-2}$	$1^{3}2^{-3}4^{3}$	$1^{3}2^{-2}4^{2}$
$1^{3}2^{-1}4^{1}$	$1^{3}2^{1}4^{-1}$	$1^{4}2^{-3}4^{2}$	$1^{4}2^{-2}4^{1}$	$1^{4}2^{0}4^{-1}$	$1^{5}2^{-3}4^{1}$
$N=4$ , $k=2$
$1^{-7}2^{17}4^{-6}$	$1^{-7}2^{18}4^{-7}$	$1^{-6}2^{14}4^{-4}$	$1^{-6}2^{15}4^{-5}$	$1^{-6}2^{16}4^{-6}$	$1^{-6}2^{17}4^{-7}$
$1^{-5}2^{11}4^{-2}$	$1^{-5}2^{12}4^{-3}$	$1^{-5}2^{13}4^{-4}$	$1^{-5}2^{14}4^{-5}$	$1^{-5}2^{15}4^{-6}$	$1^{-4}2^{9}4^{-1}$
$1^{-4}2^{10}4^{-2}$	$1^{-4}2^{11}4^{-3}$	$1^{-4}2^{12}4^{-4}$	$1^{-4}2^{13}4^{-5}$	$1^{-4}2^{14}4^{-6}$	$1^{-3}2^{5}4^{2}$
$1^{-3}2^{6}4^{1}$	$1^{-3}2^{8}4^{-1}$	$1^{-3}2^{9}4^{-2}$	$1^{-3}2^{10}4^{-3}$	$1^{-3}2^{11}4^{-4}$	$1^{-3}2^{12}4^{-5}$
$1^{-2}2^{2}4^{4}$	$1^{-2}2^{3}4^{3}$	$1^{-2}2^{4}4^{2}$	$1^{-2}2^{5}4^{1}$	$1^{-2}2^{7}4^{-1}$	$1^{-2}2^{8}4^{-2}$
$1^{-2}2^{9}4^{-3}$	$1^{-2}2^{10}4^{-4}$	$1^{-2}2^{11}4^{-5}$	$1^{-1}2^{-1}4^{6}$	$1^{-1}2^{0}4^{5}$	$1^{-1}2^{1}4^{4}$
$1^{-1}2^{2}4^{3}$	$1^{-1}2^{3}4^{2}$	$1^{-1}2^{4}4^{1}$	$1^{-1}2^{6}4^{-1}$	$1^{-1}2^{7}4^{-2}$	$1^{-1}2^{8}4^{-3}$
$1^{-1}2^{9}4^{-4}$	$1^{0}2^{-3}4^{7}$	$1^{0}2^{-2}4^{6}$	$1^{0}2^{-1}4^{5}$	$1^{0}2^{0}4^{4}$	$1^{0}2^{1}4^{3}$
$1^{0}2^{2}4^{2}$	$1^{0}2^{3}4^{1}$	$1^{0}2^{5}4^{-1}$	$1^{0}2^{6}4^{-2}$	$1^{0}2^{7}4^{-3}$	$1^{0}2^{8}4^{-4}$
$1^{1}2^{-4}4^{7}$	$1^{1}2^{-3}4^{6}$	$1^{1}2^{-2}4^{5}$	$1^{1}2^{-1}4^{4}$	$1^{1}2^{0}4^{3}$	$1^{1}2^{1}4^{2}$
$1^{1}2^{2}4^{1}$	$1^{1}2^{4}4^{-1}$	$1^{1}2^{5}4^{-2}$	$1^{1}2^{6}4^{-3}$	$1^{2}2^{-4}4^{6}$	$1^{2}2^{-3}4^{5}$
$1^{2}2^{-2}4^{4}$	$1^{2}2^{-1}4^{3}$	$1^{2}2^{0}4^{2}$	$1^{2}2^{1}4^{1}$	$1^{2}2^{3}4^{-1}$	$1^{2}2^{4}4^{-2}$
$1^{2}2^{5}4^{-3}$	$1^{3}2^{-4}4^{5}$	$1^{3}2^{-3}4^{4}$	$1^{3}2^{-2}4^{3}$	$1^{3}2^{-1}4^{2}$	$1^{3}2^{0}4^{1}$
$1^{3}2^{2}4^{-1}$	$1^{3}2^{3}4^{-2}$	$1^{4}2^{-4}4^{4}$	$1^{4}2^{-3}4^{3}$	$1^{4}2^{-2}4^{2}$	$1^{4}2^{-1}4^{1}$
$1^{4}2^{1}4^{-1}$	$1^{4}2^{2}4^{-2}$	$1^{5}2^{-4}4^{3}$	$1^{5}2^{-3}4^{2}$	$1^{5}2^{-2}4^{1}$	$1^{5}2^{0}4^{-1}$
$1^{6}2^{-4}4^{2}$	$1^{6}2^{-3}4^{1}$	$1^{6}2^{-1}4^{-1}$	$1^{7}2^{-4}4^{1}$
$N=4$ , $k=5/2$
$1^{-7}2^{16}4^{-4}$	$1^{-7}2^{17}4^{-5}$	$1^{-7}2^{18}4^{-6}$	$1^{-6}2^{13}4^{-2}$	$1^{-6}2^{14}4^{-3}$	$1^{-6}2^{15}4^{-4}$
$1^{-6}2^{16}4^{-5}$	$1^{-6}2^{17}4^{-6}$	$1^{-6}2^{18}4^{-7}$	$1^{-5}2^{11}4^{-1}$	$1^{-5}2^{12}4^{-2}$	$1^{-5}2^{13}4^{-3}$
$1^{-5}2^{14}4^{-4}$	$1^{-5}2^{15}4^{-5}$	$1^{-5}2^{16}4^{-6}$	$1^{-5}2^{17}4^{-7}$	$1^{-4}2^{7}4^{2}$	$1^{-4}2^{8}4^{1}$
$1^{-4}2^{10}4^{-1}$	$1^{-4}2^{11}4^{-2}$	$1^{-4}2^{12}4^{-3}$	$1^{-4}2^{13}4^{-4}$	$1^{-4}2^{14}4^{-5}$	$1^{-4}2^{15}4^{-6}$
$1^{-4}2^{16}4^{-7}$	$1^{-3}2^{4}4^{4}$	$1^{-3}2^{5}4^{3}$	$1^{-3}2^{6}4^{2}$	$1^{-3}2^{7}4^{1}$	$1^{-3}2^{9}4^{-1}$
$1^{-3}2^{10}4^{-2}$	$1^{-3}2^{11}4^{-3}$	$1^{-3}2^{12}4^{-4}$	$1^{-3}2^{13}4^{-5}$	$1^{-3}2^{14}4^{-6}$	$1^{-2}2^{2}4^{5}$
$1^{-2}2^{3}4^{4}$	$1^{-2}2^{4}4^{3}$	$1^{-2}2^{5}4^{2}$	$1^{-2}2^{6}4^{1}$	$1^{-2}2^{8}4^{-1}$	$1^{-2}2^{9}4^{-2}$
$1^{-2}2^{10}4^{-3}$	$1^{-2}2^{11}4^{-4}$	$1^{-2}2^{12}4^{-5}$	$1^{-2}2^{13}4^{-6}$	$1^{-1}2^{0}4^{6}$	$1^{-1}2^{1}4^{5}$
$1^{-1}2^{2}4^{4}$	$1^{-1}2^{3}4^{3}$	$1^{-1}2^{4}4^{2}$	$1^{-1}2^{5}4^{1}$	$1^{-1}2^{7}4^{-1}$	$1^{-1}2^{8}4^{-2}$
$1^{-1}2^{9}4^{-3}$	$1^{-1}2^{10}4^{-4}$	$1^{-1}2^{11}4^{-5}$	$1^{0}2^{-1}4^{6}$	$1^{0}2^{0}4^{5}$	$1^{0}2^{1}4^{4}$
$1^{0}2^{2}4^{3}$	$1^{0}2^{3}4^{2}$	$1^{0}2^{4}4^{1}$	$1^{0}2^{6}4^{-1}$	$1^{0}2^{7}4^{-2}$	$1^{0}2^{8}4^{-3}$
$1^{0}2^{9}4^{-4}$	$1^{0}2^{10}4^{-5}$	$1^{1}2^{-3}4^{7}$	$1^{1}2^{-2}4^{6}$	$1^{1}2^{-1}4^{5}$	$1^{1}2^{0}4^{4}$
$1^{1}2^{1}4^{3}$	$1^{1}2^{2}4^{2}$	$1^{1}2^{3}4^{1}$	$1^{1}2^{5}4^{-1}$	$1^{1}2^{6}4^{-2}$	$1^{1}2^{7}4^{-3}$
$1^{1}2^{8}4^{-4}$	$1^{2}2^{-4}4^{7}$	$1^{2}2^{-3}4^{6}$	$1^{2}2^{-2}4^{5}$	$1^{2}2^{-1}4^{4}$	$1^{2}2^{0}4^{3}$
$1^{2}2^{1}4^{2}$	$1^{2}2^{2}4^{1}$	$1^{2}2^{4}4^{-1}$	$1^{2}2^{5}4^{-2}$	$1^{2}2^{6}4^{-3}$	$1^{2}2^{7}4^{-4}$
$1^{3}2^{-5}4^{7}$	$1^{3}2^{-4}4^{6}$	$1^{3}2^{-3}4^{5}$	$1^{3}2^{-2}4^{4}$	$1^{3}2^{-1}4^{3}$	$1^{3}2^{0}4^{2}$
$1^{3}2^{1}4^{1}$	$1^{3}2^{3}4^{-1}$	$1^{3}2^{4}4^{-2}$	$1^{3}2^{5}4^{-3}$	$1^{4}2^{-5}4^{6}$	$1^{4}2^{-4}4^{5}$
$1^{4}2^{-3}4^{4}$	$1^{4}2^{-2}4^{3}$	$1^{4}2^{-1}4^{2}$	$1^{4}2^{0}4^{1}$	$1^{4}2^{2}4^{-1}$	$1^{4}2^{3}4^{-2}$
$1^{4}2^{4}4^{-3}$	$1^{5}2^{-5}4^{5}$	$1^{5}2^{-4}4^{4}$	$1^{5}2^{-3}4^{3}$	$1^{5}2^{-2}4^{2}$	$1^{5}2^{-1}4^{1}$
$1^{5}2^{1}4^{-1}$	$1^{5}2^{2}4^{-2}$	$1^{6}2^{-5}4^{4}$	$1^{6}2^{-4}4^{3}$	$1^{6}2^{-3}4^{2}$	$1^{6}2^{-2}4^{1}$
$1^{6}2^{0}4^{-1}$	$1^{7}2^{-5}4^{3}$	$1^{7}2^{-4}4^{2}$	$1^{7}2^{-3}4^{1}$
$N=4$ , $k=3$
$1^{-7}2^{15}4^{-2}$	$1^{-7}2^{16}4^{-3}$	$1^{-7}2^{17}4^{-4}$	$1^{-6}2^{13}4^{-1}$	$1^{-6}2^{14}4^{-2}$	$1^{-6}2^{15}4^{-3}$
$1^{-6}2^{16}4^{-4}$	$1^{-6}2^{17}4^{-5}$	$1^{-5}2^{9}4^{2}$	$1^{-5}2^{10}4^{1}$	$1^{-5}2^{12}4^{-1}$	$1^{-5}2^{13}4^{-2}$
$1^{-5}2^{14}4^{-3}$	$1^{-5}2^{15}4^{-4}$	$1^{-5}2^{16}4^{-5}$	$1^{-5}2^{17}4^{-6}$	$1^{-4}2^{7}4^{3}$	$1^{-4}2^{8}4^{2}$
$1^{-4}2^{9}4^{1}$	$1^{-4}2^{11}4^{-1}$	$1^{-4}2^{12}4^{-2}$	$1^{-4}2^{13}4^{-3}$	$1^{-4}2^{14}4^{-4}$	$1^{-4}2^{15}4^{-5}$
$1^{-4}2^{16}4^{-6}$	$1^{-4}2^{17}4^{-7}$	$1^{-3}2^{5}4^{4}$	$1^{-3}2^{6}4^{3}$	$1^{-3}2^{7}4^{2}$	$1^{-3}2^{8}4^{1}$
$1^{-3}2^{10}4^{-1}$	$1^{-3}2^{11}4^{-2}$	$1^{-3}2^{12}4^{-3}$	$1^{-3}2^{13}4^{-4}$	$1^{-3}2^{14}4^{-5}$	$1^{-3}2^{15}4^{-6}$
$1^{-3}2^{16}4^{-7}$	$1^{-2}2^{4}4^{4}$	$1^{-2}2^{5}4^{3}$	$1^{-2}2^{6}4^{2}$	$1^{-2}2^{7}4^{1}$	$1^{-2}2^{9}4^{-1}$
$1^{-2}2^{10}4^{-2}$	$1^{-2}2^{11}4^{-3}$	$1^{-2}2^{12}4^{-4}$	$1^{-2}2^{13}4^{-5}$	$1^{-2}2^{14}4^{-6}$	$1^{-2}2^{15}4^{-7}$
$1^{-1}2^{2}4^{5}$	$1^{-1}2^{3}4^{4}$	$1^{-1}2^{4}4^{3}$	$1^{-1}2^{5}4^{2}$	$1^{-1}2^{6}4^{1}$	$1^{-1}2^{8}4^{-1}$
$1^{-1}2^{9}4^{-2}$	$1^{-1}2^{10}4^{-3}$	$1^{-1}2^{11}4^{-4}$	$1^{-1}2^{12}4^{-5}$	$1^{-1}2^{13}4^{-6}$	$1^{0}2^{1}4^{5}$
$1^{0}2^{2}4^{4}$	$1^{0}2^{3}4^{3}$	$1^{0}2^{4}4^{2}$	$1^{0}2^{5}4^{1}$	$1^{0}2^{7}4^{-1}$	$1^{0}2^{8}4^{-2}$
$1^{0}2^{9}4^{-3}$	$1^{0}2^{10}4^{-4}$	$1^{0}2^{11}4^{-5}$	$1^{0}2^{12}4^{-6}$	$1^{1}2^{-1}4^{6}$	$1^{1}2^{0}4^{5}$
$1^{1}2^{1}4^{4}$	$1^{1}2^{2}4^{3}$	$1^{1}2^{3}4^{2}$	$1^{1}2^{4}4^{1}$	$1^{1}2^{6}4^{-1}$	$1^{1}2^{7}4^{-2}$
$1^{1}2^{8}4^{-3}$	$1^{1}2^{9}4^{-4}$	$1^{1}2^{10}4^{-5}$	$1^{2}2^{-2}4^{6}$	$1^{2}2^{-1}4^{5}$	$1^{2}2^{0}4^{4}$
$1^{2}2^{1}4^{3}$	$1^{2}2^{2}4^{2}$	$1^{2}2^{3}4^{1}$	$1^{2}2^{5}4^{-1}$	$1^{2}2^{6}4^{-2}$	$1^{2}2^{7}4^{-3}$
$1^{2}2^{8}4^{-4}$	$1^{2}2^{9}4^{-5}$	$1^{3}2^{-4}4^{7}$	$1^{3}2^{-3}4^{6}$	$1^{3}2^{-2}4^{5}$	$1^{3}2^{-1}4^{4}$
$1^{3}2^{0}4^{3}$	$1^{3}2^{1}4^{2}$	$1^{3}2^{2}4^{1}$	$1^{3}2^{4}4^{-1}$	$1^{3}2^{5}4^{-2}$	$1^{3}2^{6}4^{-3}$
$1^{3}2^{7}4^{-4}$	$1^{4}2^{-5}4^{7}$	$1^{4}2^{-4}4^{6}$	$1^{4}2^{-3}4^{5}$	$1^{4}2^{-2}4^{4}$	$1^{4}2^{-1}4^{3}$
$1^{4}2^{0}4^{2}$	$1^{4}2^{1}4^{1}$	$1^{4}2^{3}4^{-1}$	$1^{4}2^{4}4^{-2}$	$1^{4}2^{5}4^{-3}$	$1^{5}2^{-6}4^{7}$
$1^{5}2^{-5}4^{6}$	$1^{5}2^{-4}4^{5}$	$1^{5}2^{-3}4^{4}$	$1^{5}2^{-2}4^{3}$	$1^{5}2^{-1}4^{2}$	$1^{5}2^{0}4^{1}$
$1^{5}2^{2}4^{-1}$	$1^{6}2^{-6}4^{6}$	$1^{6}2^{-5}4^{5}$	$1^{6}2^{-4}4^{4}$	$1^{6}2^{-3}4^{3}$	$1^{6}2^{-2}4^{2}$
$1^{6}2^{-1}4^{1}$	$1^{7}2^{-6}4^{5}$	$1^{7}2^{-5}4^{4}$	$1^{7}2^{-4}4^{3}$
$N=4$ , $k=7/2$
$1^{-7}2^{15}4^{-1}$	$1^{-7}2^{16}4^{-2}$	$1^{-6}2^{12}4^{1}$	$1^{-6}2^{14}4^{-1}$	$1^{-6}2^{15}4^{-2}$	$1^{-6}2^{16}4^{-3}$
$1^{-5}2^{10}4^{2}$	$1^{-5}2^{11}4^{1}$	$1^{-5}2^{13}4^{-1}$	$1^{-5}2^{14}4^{-2}$	$1^{-5}2^{15}4^{-3}$	$1^{-5}2^{16}4^{-4}$
$1^{-4}2^{9}4^{2}$	$1^{-4}2^{10}4^{1}$	$1^{-4}2^{12}4^{-1}$	$1^{-4}2^{13}4^{-2}$	$1^{-4}2^{14}4^{-3}$	$1^{-4}2^{15}4^{-4}$
$1^{-4}2^{16}4^{-5}$	$1^{-3}2^{7}4^{3}$	$1^{-3}2^{8}4^{2}$	$1^{-3}2^{9}4^{1}$	$1^{-3}2^{11}4^{-1}$	$1^{-3}2^{12}4^{-2}$
$1^{-3}2^{13}4^{-3}$	$1^{-3}2^{14}4^{-4}$	$1^{-3}2^{15}4^{-5}$	$1^{-3}2^{16}4^{-6}$	$1^{-2}2^{6}4^{3}$	$1^{-2}2^{7}4^{2}$
$1^{-2}2^{8}4^{1}$	$1^{-2}2^{10}4^{-1}$	$1^{-2}2^{11}4^{-2}$	$1^{-2}2^{12}4^{-3}$	$1^{-2}2^{13}4^{-4}$	$1^{-2}2^{14}4^{-5}$
$1^{-2}2^{15}4^{-6}$	$1^{-2}2^{16}4^{-7}$	$1^{-1}2^{4}4^{4}$	$1^{-1}2^{5}4^{3}$	$1^{-1}2^{6}4^{2}$	$1^{-1}2^{7}4^{1}$
$1^{-1}2^{9}4^{-1}$	$1^{-1}2^{10}4^{-2}$	$1^{-1}2^{11}4^{-3}$	$1^{-1}2^{12}4^{-4}$	$1^{-1}2^{13}4^{-5}$	$1^{-1}2^{14}4^{-6}$
$1^{-1}2^{15}4^{-7}$	$1^{0}2^{3}4^{4}$	$1^{0}2^{4}4^{3}$	$1^{0}2^{5}4^{2}$	$1^{0}2^{6}4^{1}$	$1^{0}2^{8}4^{-1}$
$1^{0}2^{9}4^{-2}$	$1^{0}2^{10}4^{-3}$	$1^{0}2^{11}4^{-4}$	$1^{0}2^{12}4^{-5}$	$1^{0}2^{13}4^{-6}$	$1^{0}2^{14}4^{-7}$
$1^{1}2^{1}4^{5}$	$1^{1}2^{2}4^{4}$	$1^{1}2^{3}4^{3}$	$1^{1}2^{4}4^{2}$	$1^{1}2^{5}4^{1}$	$1^{1}2^{7}4^{-1}$
$1^{1}2^{8}4^{-2}$	$1^{1}2^{9}4^{-3}$	$1^{1}2^{10}4^{-4}$	$1^{1}2^{11}4^{-5}$	$1^{1}2^{12}4^{-6}$	$1^{2}2^{0}4^{5}$
$1^{2}2^{1}4^{4}$	$1^{2}2^{2}4^{3}$	$1^{2}2^{3}4^{2}$	$1^{2}2^{4}4^{1}$	$1^{2}2^{6}4^{-1}$	$1^{2}2^{7}4^{-2}$
$1^{2}2^{8}4^{-3}$	$1^{2}2^{9}4^{-4}$	$1^{2}2^{10}4^{-5}$	$1^{3}2^{-2}4^{6}$	$1^{3}2^{-1}4^{5}$	$1^{3}2^{0}4^{4}$
$1^{3}2^{1}4^{3}$	$1^{3}2^{2}4^{2}$	$1^{3}2^{3}4^{1}$	$1^{3}2^{5}4^{-1}$	$1^{3}2^{6}4^{-2}$	$1^{3}2^{7}4^{-3}$
$1^{4}2^{-3}4^{6}$	$1^{4}2^{-2}4^{5}$	$1^{4}2^{-1}4^{4}$	$1^{4}2^{0}4^{3}$	$1^{4}2^{1}4^{2}$	$1^{4}2^{2}4^{1}$
$1^{4}2^{4}4^{-1}$	$1^{5}2^{-5}4^{7}$	$1^{5}2^{-4}4^{6}$	$1^{5}2^{-3}4^{5}$	$1^{5}2^{-2}4^{4}$	$1^{5}2^{-1}4^{3}$
$1^{5}2^{0}4^{2}$	$1^{5}2^{1}4^{1}$	$1^{6}2^{-6}4^{7}$	$1^{6}2^{-5}4^{6}$	$1^{6}2^{-4}4^{5}$	$1^{6}2^{-3}4^{4}$
$1^{6}2^{-2}4^{3}$	$1^{7}2^{-7}4^{7}$	$1^{7}2^{-6}4^{6}$	$1^{7}2^{-5}4^{5}$
$N=4$ , $k=4$
$1^{-6}2^{15}4^{-1}$	$1^{-5}2^{12}4^{1}$	$1^{-5}2^{14}4^{-1}$	$1^{-5}2^{15}4^{-2}$	$1^{-4}2^{11}4^{1}$	$1^{-4}2^{13}4^{-1}$
$1^{-4}2^{14}4^{-2}$	$1^{-4}2^{15}4^{-3}$	$1^{-3}2^{9}4^{2}$	$1^{-3}2^{10}4^{1}$	$1^{-3}2^{12}4^{-1}$	$1^{-3}2^{13}4^{-2}$
$1^{-3}2^{14}4^{-3}$	$1^{-3}2^{15}4^{-4}$	$1^{-2}2^{8}4^{2}$	$1^{-2}2^{9}4^{1}$	$1^{-2}2^{11}4^{-1}$	$1^{-2}2^{12}4^{-2}$
$1^{-2}2^{13}4^{-3}$	$1^{-2}2^{14}4^{-4}$	$1^{-2}2^{15}4^{-5}$	$1^{-1}2^{6}4^{3}$	$1^{-1}2^{7}4^{2}$	$1^{-1}2^{8}4^{1}$
$1^{-1}2^{10}4^{-1}$	$1^{-1}2^{11}4^{-2}$	$1^{-1}2^{12}4^{-3}$	$1^{-1}2^{13}4^{-4}$	$1^{-1}2^{14}4^{-5}$	$1^{-1}2^{15}4^{-6}$
$1^{0}2^{5}4^{3}$	$1^{0}2^{6}4^{2}$	$1^{0}2^{7}4^{1}$	$1^{0}2^{9}4^{-1}$	$1^{0}2^{10}4^{-2}$	$1^{0}2^{11}4^{-3}$
$1^{0}2^{12}4^{-4}$	$1^{0}2^{13}4^{-5}$	$1^{0}2^{14}4^{-6}$	$1^{0}2^{15}4^{-7}$	$1^{1}2^{3}4^{4}$	$1^{1}2^{4}4^{3}$
$1^{1}2^{5}4^{2}$	$1^{1}2^{6}4^{1}$	$1^{1}2^{8}4^{-1}$	$1^{1}2^{9}4^{-2}$	$1^{1}2^{10}4^{-3}$	$1^{1}2^{11}4^{-4}$
$1^{1}2^{12}4^{-5}$	$1^{2}2^{2}4^{4}$	$1^{2}2^{3}4^{3}$	$1^{2}2^{4}4^{2}$	$1^{2}2^{5}4^{1}$	$1^{2}2^{7}4^{-1}$
$1^{2}2^{8}4^{-2}$	$1^{2}2^{9}4^{-3}$	$1^{3}2^{0}4^{5}$	$1^{3}2^{1}4^{4}$	$1^{3}2^{2}4^{3}$	$1^{3}2^{3}4^{2}$
$1^{3}2^{4}4^{1}$	$1^{3}2^{6}4^{-1}$	$1^{4}2^{-1}4^{5}$	$1^{4}2^{0}4^{4}$	$1^{4}2^{1}4^{3}$	$1^{4}2^{2}4^{2}$
$1^{4}2^{3}4^{1}$	$1^{5}2^{-3}4^{6}$	$1^{5}2^{-2}4^{5}$	$1^{5}2^{-1}4^{4}$	$1^{5}2^{0}4^{3}$	$1^{6}2^{-4}4^{6}$
$1^{6}2^{-3}4^{5}$	$1^{7}2^{-6}4^{7}$
$N=4$ , $k=9/2$
$1^{-4}2^{14}4^{-1}$	$1^{-3}2^{11}4^{1}$	$1^{-3}2^{13}4^{-1}$	$1^{-3}2^{14}4^{-2}$	$1^{-2}2^{10}4^{1}$	$1^{-2}2^{12}4^{-1}$
$1^{-2}2^{13}4^{-2}$	$1^{-2}2^{14}4^{-3}$	$1^{-1}2^{8}4^{2}$	$1^{-1}2^{9}4^{1}$	$1^{-1}2^{11}4^{-1}$	$1^{-1}2^{12}4^{-2}$
$1^{-1}2^{13}4^{-3}$	$1^{-1}2^{14}4^{-4}$	$1^{0}2^{7}4^{2}$	$1^{0}2^{8}4^{1}$	$1^{0}2^{10}4^{-1}$	$1^{0}2^{11}4^{-2}$
$1^{0}2^{12}4^{-3}$	$1^{0}2^{13}4^{-4}$	$1^{0}2^{14}4^{-5}$	$1^{1}2^{5}4^{3}$	$1^{1}2^{6}4^{2}$	$1^{1}2^{7}4^{1}$
$1^{1}2^{9}4^{-1}$	$1^{1}2^{10}4^{-2}$	$1^{1}2^{11}4^{-3}$	$1^{2}2^{4}4^{3}$	$1^{2}2^{5}4^{2}$	$1^{2}2^{6}4^{1}$
$1^{2}2^{8}4^{-1}$	$1^{3}2^{2}4^{4}$	$1^{3}2^{3}4^{3}$	$1^{3}2^{4}4^{2}$	$1^{3}2^{5}4^{1}$	$1^{4}2^{1}4^{4}$
$1^{4}2^{2}4^{3}$	$1^{5}2^{-1}4^{5}$
$N=4$ , $k=5$
$1^{-2}2^{13}4^{-1}$	$1^{-1}2^{10}4^{1}$	$1^{-1}2^{12}4^{-1}$	$1^{-1}2^{13}4^{-2}$	$1^{0}2^{9}4^{1}$	$1^{0}2^{11}4^{-1}$
$1^{0}2^{12}4^{-2}$	$1^{0}2^{13}4^{-3}$	$1^{1}2^{7}4^{2}$	$1^{1}2^{8}4^{1}$	$1^{1}2^{10}4^{-1}$	$1^{2}2^{6}4^{2}$
$1^{2}2^{7}4^{1}$	$1^{3}2^{4}4^{3}$
$N=4$ , $k=11/2$
$1^{0}2^{12}4^{-1}$	$1^{1}2^{9}4^{1}$
$N=5$ , $k=1/2$
$1^{0}5^{1}$
$N=5$ , $k=1$
$1^{0}5^{2}$	$1^{1}5^{1}$
$N=5$ , $k=3/2$
$1^{0}5^{3}$	$1^{1}5^{2}$	$1^{2}5^{1}$
$N=5$ , $k=2$
$1^{0}5^{4}$	$1^{1}5^{3}$	$1^{2}5^{2}$	$1^{3}5^{1}$
$N=5$ , $k=5/2$
$1^{1}5^{4}$	$1^{2}5^{3}$	$1^{3}5^{2}$	$1^{4}5^{1}$
$N=5$ , $k=3$
$1^{2}5^{4}$	$1^{3}5^{3}$	$1^{4}5^{2}$
$N=5$ , $k=7/2$
$1^{3}5^{4}$	$1^{4}5^{3}$
$N=6$ , $k=1/2$
$1^{-1}2^{1}3^{2}6^{-1}$	$1^{-1}2^{2}3^{1}6^{-1}$	$1^{0}2^{0}3^{-1}6^{2}$	$1^{0}2^{0}3^{0}6^{1}$	$1^{0}2^{0}3^{2}6^{-1}$	$1^{1}2^{-1}3^{-1}6^{2}$
$1^{2}2^{-1}3^{-1}6^{1}$
$N=6$ , $k=1$
$1^{-2}2^{2}3^{4}6^{-2}$	$1^{-2}2^{3}3^{2}6^{-1}$	$1^{-2}2^{3}3^{3}6^{-2}$	$1^{-2}2^{4}3^{1}6^{-1}$	$1^{-2}2^{4}3^{2}6^{-2}$	$1^{-1}2^{1}3^{1}6^{1}$
$1^{-1}2^{1}3^{2}6^{0}$	$1^{-1}2^{1}3^{3}6^{-1}$	$1^{-1}2^{1}3^{4}6^{-2}$	$1^{-1}2^{2}3^{-1}6^{2}$	$1^{-1}2^{2}3^{0}6^{1}$	$1^{-1}2^{2}3^{1}6^{0}$
$1^{-1}2^{2}3^{2}6^{-1}$	$1^{-1}2^{2}3^{3}6^{-2}$	$1^{-1}2^{3}3^{1}6^{-1}$	$1^{0}2^{-1}3^{0}6^{3}$	$1^{0}2^{0}3^{-2}6^{4}$	$1^{0}2^{0}3^{-1}6^{3}$
$1^{0}2^{0}3^{0}6^{2}$	$1^{0}2^{0}3^{1}6^{1}$	$1^{0}2^{0}3^{3}6^{-1}$	$1^{0}2^{0}3^{4}6^{-2}$	$1^{0}2^{1}3^{-1}6^{2}$	$1^{0}2^{1}3^{0}6^{1}$
$1^{0}2^{1}3^{1}6^{0}$	$1^{0}2^{1}3^{2}6^{-1}$	$1^{0}2^{2}3^{1}6^{-1}$	$1^{0}2^{3}3^{0}6^{-1}$	$1^{1}2^{-1}3^{-2}6^{4}$	$1^{1}2^{-1}3^{-1}6^{3}$
$1^{1}2^{-1}3^{0}6^{2}$	$1^{1}2^{-1}3^{1}6^{1}$	$1^{1}2^{0}3^{-1}6^{2}$	$1^{1}2^{0}3^{0}6^{1}$	$1^{1}2^{0}3^{2}6^{-1}$	$1^{1}2^{1}3^{-1}6^{1}$
$1^{1}2^{1}3^{1}6^{-1}$	$1^{2}2^{-2}3^{-2}6^{4}$	$1^{2}2^{-1}3^{-2}6^{3}$	$1^{2}2^{-1}3^{-1}6^{2}$	$1^{2}2^{-1}3^{0}6^{1}$	$1^{2}2^{-1}3^{1}6^{0}$
$1^{2}2^{-1}3^{2}6^{-1}$	$1^{2}2^{0}3^{-1}6^{1}$	$1^{3}2^{-2}3^{-2}6^{3}$	$1^{3}2^{-2}3^{-1}6^{2}$	$1^{3}2^{-1}3^{-1}6^{1}$	$1^{4}2^{-2}3^{-2}6^{2}$
$1^{4}2^{-2}3^{-1}6^{1}$
$N=6$ , $k=3/2$
$1^{-3}2^{4}3^{4}6^{-2}$	$1^{-3}2^{4}3^{5}6^{-3}$	$1^{-3}2^{5}3^{2}6^{-1}$	$1^{-3}2^{5}3^{3}6^{-2}$	$1^{-3}2^{5}3^{4}6^{-3}$	$1^{-2}2^{2}3^{3}6^{0}$
$1^{-2}2^{2}3^{4}6^{-1}$	$1^{-2}2^{2}3^{5}6^{-2}$	$1^{-2}2^{3}3^{1}6^{1}$	$1^{-2}2^{3}3^{2}6^{0}$	$1^{-2}2^{3}3^{3}6^{-1}$	$1^{-2}2^{3}3^{4}6^{-2}$
$1^{-2}2^{3}3^{5}6^{-3}$	$1^{-2}2^{4}3^{-1}6^{2}$	$1^{-2}2^{4}3^{0}6^{1}$	$1^{-2}2^{4}3^{1}6^{0}$	$1^{-2}2^{4}3^{2}6^{-1}$	$1^{-2}2^{4}3^{3}6^{-2}$
$1^{-2}2^{4}3^{4}6^{-3}$	$1^{-2}2^{5}3^{1}6^{-1}$	$1^{-2}2^{5}3^{2}6^{-2}$	$1^{-1}2^{0}3^{2}6^{2}$	$1^{-1}2^{0}3^{3}6^{1}$	$1^{-1}2^{1}3^{0}6^{3}$
$1^{-1}2^{1}3^{1}6^{2}$	$1^{-1}2^{1}3^{2}6^{1}$	$1^{-1}2^{1}3^{3}6^{0}$	$1^{-1}2^{1}3^{4}6^{-1}$	$1^{-1}2^{1}3^{5}6^{-2}$	$1^{-1}2^{2}3^{-2}6^{4}$
$1^{-1}2^{2}3^{-1}6^{3}$	$1^{-1}2^{2}3^{0}6^{2}$	$1^{-1}2^{2}3^{1}6^{1}$	$1^{-1}2^{2}3^{2}6^{0}$	$1^{-1}2^{2}3^{3}6^{-1}$	$1^{-1}2^{2}3^{4}6^{-2}$
$1^{-1}2^{2}3^{5}6^{-3}$	$1^{-1}2^{3}3^{-1}6^{2}$	$1^{-1}2^{3}3^{0}6^{1}$	$1^{-1}2^{3}3^{1}6^{0}$	$1^{-1}2^{3}3^{2}6^{-1}$	$1^{-1}2^{3}3^{3}6^{-2}$
$1^{-1}2^{4}3^{1}6^{-1}$	$1^{-1}2^{4}3^{2}6^{-2}$	$1^{0}2^{-1}3^{0}6^{4}$	$1^{0}2^{-1}3^{1}6^{3}$	$1^{0}2^{-1}3^{2}6^{2}$	$1^{0}2^{0}3^{-1}6^{4}$
$1^{0}2^{0}3^{0}6^{3}$	$1^{0}2^{0}3^{1}6^{2}$	$1^{0}2^{0}3^{2}6^{1}$	$1^{0}2^{0}3^{4}6^{-1}$	$1^{0}2^{1}3^{-2}6^{4}$	$1^{0}2^{1}3^{-1}6^{3}$
$1^{0}2^{1}3^{0}6^{2}$	$1^{0}2^{1}3^{1}6^{1}$	$1^{0}2^{1}3^{2}6^{0}$	$1^{0}2^{1}3^{3}6^{-1}$	$1^{0}2^{1}3^{4}6^{-2}$	$1^{0}2^{2}3^{-1}6^{2}$
$1^{0}2^{2}3^{0}6^{1}$	$1^{0}2^{2}3^{1}6^{0}$	$1^{0}2^{2}3^{2}6^{-1}$	$1^{0}2^{2}3^{3}6^{-2}$	$1^{0}2^{3}3^{-1}6^{1}$	$1^{0}2^{3}3^{1}6^{-1}$
$1^{0}2^{3}3^{2}6^{-2}$	$1^{0}2^{4}3^{0}6^{-1}$	$1^{1}2^{-1}3^{-2}6^{5}$	$1^{1}2^{-1}3^{-1}6^{4}$	$1^{1}2^{-1}3^{0}6^{3}$	$1^{1}2^{-1}3^{1}6^{2}$
$1^{1}2^{-1}3^{2}6^{1}$	$1^{1}2^{-1}3^{3}6^{0}$	$1^{1}2^{0}3^{-2}6^{4}$	$1^{1}2^{0}3^{-1}6^{3}$	$1^{1}2^{0}3^{0}6^{2}$	$1^{1}2^{0}3^{1}6^{1}$
$1^{1}2^{0}3^{3}6^{-1}$	$1^{1}2^{0}3^{4}6^{-2}$	$1^{1}2^{1}3^{-2}6^{3}$	$1^{1}2^{1}3^{-1}6^{2}$	$1^{1}2^{1}3^{0}6^{1}$	$1^{1}2^{1}3^{1}6^{0}$
$1^{1}2^{1}3^{2}6^{-1}$	$1^{1}2^{1}3^{3}6^{-2}$	$1^{1}2^{2}3^{-1}6^{1}$	$1^{1}2^{2}3^{1}6^{-1}$	$1^{1}2^{3}3^{0}6^{-1}$	$1^{2}2^{-2}3^{-2}6^{5}$
$1^{2}2^{-2}3^{-1}6^{4}$	$1^{2}2^{-2}3^{0}6^{3}$	$1^{2}2^{-1}3^{-3}6^{5}$	$1^{2}2^{-1}3^{-2}6^{4}$	$1^{2}2^{-1}3^{-1}6^{3}$	$1^{2}2^{-1}3^{0}6^{2}$
$1^{2}2^{-1}3^{1}6^{1}$	$1^{2}2^{-1}3^{2}6^{0}$	$1^{2}2^{-1}3^{3}6^{-1}$	$1^{2}2^{-1}3^{4}6^{-2}$	$1^{2}2^{0}3^{-2}6^{3}$	$1^{2}2^{0}3^{-1}6^{2}$
$1^{2}2^{0}3^{0}6^{1}$	$1^{2}2^{0}3^{2}6^{-1}$	$1^{2}2^{1}3^{-1}6^{1}$	$1^{2}2^{1}3^{1}6^{-1}$	$1^{2}2^{2}3^{-1}6^{0}$	$1^{2}2^{2}3^{0}6^{-1}$
$1^{3}2^{-2}3^{-3}6^{5}$	$1^{3}2^{-2}3^{-2}6^{4}$	$1^{3}2^{-2}3^{-1}6^{3}$	$1^{3}2^{-2}3^{0}6^{2}$	$1^{3}2^{-2}3^{1}6^{1}$	$1^{3}2^{-1}3^{-2}6^{3}$
$1^{3}2^{-1}3^{-1}6^{2}$	$1^{3}2^{-1}3^{0}6^{1}$	$1^{3}2^{-1}3^{1}6^{0}$	$1^{3}2^{-1}3^{2}6^{-1}$	$1^{3}2^{0}3^{-2}6^{2}$	$1^{3}2^{0}3^{-1}6^{1}$
$1^{3}2^{0}3^{1}6^{-1}$	$1^{3}2^{1}3^{-1}6^{0}$	$1^{4}2^{-3}3^{-3}6^{5}$	$1^{4}2^{-3}3^{-2}6^{4}$	$1^{4}2^{-2}3^{-3}6^{4}$	$1^{4}2^{-2}3^{-2}6^{3}$
$1^{4}2^{-2}3^{-1}6^{2}$	$1^{4}2^{-2}3^{0}6^{1}$	$1^{4}2^{-2}3^{1}6^{0}$	$1^{4}2^{-2}3^{2}6^{-1}$	$1^{4}2^{-1}3^{-2}6^{2}$	$1^{4}2^{-1}3^{-1}6^{1}$
$1^{5}2^{-3}3^{-3}6^{4}$	$1^{5}2^{-3}3^{-2}6^{3}$	$1^{5}2^{-3}3^{-1}6^{2}$	$1^{5}2^{-2}3^{-2}6^{2}$	$1^{5}2^{-2}3^{-1}6^{1}$
$N=6$ , $k=2$
$1^{-4}2^{6}3^{5}6^{-3}$	$1^{-4}2^{6}3^{6}6^{-4}$	$1^{-3}2^{4}3^{3}6^{0}$	$1^{-3}2^{4}3^{4}6^{-1}$	$1^{-3}2^{4}3^{5}6^{-2}$	$1^{-3}2^{5}3^{2}6^{0}$
$1^{-3}2^{5}3^{3}6^{-1}$	$1^{-3}2^{5}3^{4}6^{-2}$	$1^{-3}2^{5}3^{5}6^{-3}$	$1^{-3}2^{5}3^{6}6^{-4}$	$1^{-2}2^{2}3^{2}6^{2}$	$1^{-2}2^{2}3^{3}6^{1}$
$1^{-2}2^{2}3^{4}6^{0}$	$1^{-2}2^{3}3^{0}6^{3}$	$1^{-2}2^{3}3^{1}6^{2}$	$1^{-2}2^{3}3^{2}6^{1}$	$1^{-2}2^{3}3^{3}6^{0}$	$1^{-2}2^{3}3^{4}6^{-1}$
$1^{-2}2^{3}3^{5}6^{-2}$	$1^{-2}2^{4}3^{-1}6^{3}$	$1^{-2}2^{4}3^{0}6^{2}$	$1^{-2}2^{4}3^{1}6^{1}$	$1^{-2}2^{4}3^{2}6^{0}$	$1^{-2}2^{4}3^{3}6^{-1}$
$1^{-2}2^{4}3^{4}6^{-2}$	$1^{-2}2^{4}3^{5}6^{-3}$	$1^{-2}2^{5}3^{3}6^{-2}$	$1^{-2}2^{5}3^{4}6^{-3}$	$1^{-1}2^{0}3^{2}6^{3}$	$1^{-1}2^{0}3^{3}6^{2}$
$1^{-1}2^{1}3^{1}6^{3}$	$1^{-1}2^{1}3^{2}6^{2}$	$1^{-1}2^{1}3^{3}6^{1}$	$1^{-1}2^{1}3^{4}6^{0}$	$1^{-1}2^{2}3^{0}6^{3}$	$1^{-1}2^{2}3^{1}6^{2}$
$1^{-1}2^{2}3^{2}6^{1}$	$1^{-1}2^{2}3^{3}6^{0}$	$1^{-1}2^{2}3^{4}6^{-1}$	$1^{-1}2^{3}3^{-2}6^{4}$	$1^{-1}2^{3}3^{-1}6^{3}$	$1^{-1}2^{3}3^{0}6^{2}$
$1^{-1}2^{3}3^{1}6^{1}$	$1^{-1}2^{3}3^{2}6^{0}$	$1^{-1}2^{3}3^{3}6^{-1}$	$1^{-1}2^{3}3^{4}6^{-2}$	$1^{-1}2^{3}3^{5}6^{-3}$	$1^{-1}2^{4}3^{0}6^{1}$
$1^{-1}2^{4}3^{1}6^{0}$	$1^{-1}2^{4}3^{2}6^{-1}$	$1^{-1}2^{4}3^{3}6^{-2}$	$1^{-1}2^{4}3^{4}6^{-3}$	$1^{0}2^{-1}3^{2}6^{3}$	$1^{0}2^{-1}3^{3}6^{2}$
$1^{0}2^{0}3^{1}6^{3}$	$1^{0}2^{0}3^{2}6^{2}$	$1^{0}2^{0}3^{3}6^{1}$	$1^{0}2^{1}3^{-1}6^{4}$	$1^{0}2^{1}3^{0}6^{3}$	$1^{0}2^{1}3^{1}6^{2}$
$1^{0}2^{1}3^{2}6^{1}$	$1^{0}2^{1}3^{3}6^{0}$	$1^{0}2^{1}3^{4}6^{-1}$	$1^{0}2^{2}3^{-2}6^{4}$	$1^{0}2^{2}3^{-1}6^{3}$	$1^{0}2^{2}3^{0}6^{2}$
$1^{0}2^{2}3^{1}6^{1}$	$1^{0}2^{2}3^{2}6^{0}$	$1^{0}2^{2}3^{3}6^{-1}$	$1^{0}2^{2}3^{4}6^{-2}$	$1^{0}2^{2}3^{5}6^{-3}$	$1^{0}2^{3}3^{-2}6^{3}$
$1^{0}2^{3}3^{-1}6^{2}$	$1^{0}2^{3}3^{0}6^{1}$	$1^{0}2^{3}3^{1}6^{0}$	$1^{0}2^{3}3^{2}6^{-1}$	$1^{0}2^{3}3^{3}6^{-2}$	$1^{0}2^{3}3^{4}6^{-3}$
$1^{0}2^{4}3^{1}6^{-1}$	$1^{0}2^{4}3^{2}6^{-2}$	$1^{1}2^{-1}3^{0}6^{4}$	$1^{1}2^{-1}3^{1}6^{3}$	$1^{1}2^{-1}3^{2}6^{2}$	$1^{1}2^{-1}3^{3}6^{1}$
$1^{1}2^{0}3^{-1}6^{4}$	$1^{1}2^{0}3^{0}6^{3}$	$1^{1}2^{0}3^{1}6^{2}$	$1^{1}2^{0}3^{2}6^{1}$	$1^{1}2^{0}3^{4}6^{-1}$	$1^{1}2^{1}3^{-2}6^{4}$
$1^{1}2^{1}3^{-1}6^{3}$	$1^{1}2^{1}3^{0}6^{2}$	$1^{1}2^{1}3^{1}6^{1}$	$1^{1}2^{1}3^{2}6^{0}$	$1^{1}2^{1}3^{3}6^{-1}$	$1^{1}2^{1}3^{4}6^{-2}$
$1^{1}2^{2}3^{-2}6^{3}$	$1^{1}2^{2}3^{-1}6^{2}$	$1^{1}2^{2}3^{0}6^{1}$	$1^{1}2^{2}3^{1}6^{0}$	$1^{1}2^{2}3^{2}6^{-1}$	$1^{1}2^{2}3^{3}6^{-2}$
$1^{1}2^{3}3^{-1}6^{1}$	$1^{1}2^{3}3^{1}6^{-1}$	$1^{1}2^{3}3^{2}6^{-2}$	$1^{2}2^{-2}3^{0}6^{4}$	$1^{2}2^{-2}3^{1}6^{3}$	$1^{2}2^{-2}3^{2}6^{2}$
$1^{2}2^{-1}3^{-1}6^{4}$	$1^{2}2^{-1}3^{0}6^{3}$	$1^{2}2^{-1}3^{1}6^{2}$	$1^{2}2^{-1}3^{2}6^{1}$	$1^{2}2^{-1}3^{3}6^{0}$	$1^{2}2^{0}3^{-3}6^{5}$
$1^{2}2^{0}3^{-2}6^{4}$	$1^{2}2^{0}3^{-1}6^{3}$	$1^{2}2^{0}3^{0}6^{2}$	$1^{2}2^{0}3^{1}6^{1}$	$1^{2}2^{0}3^{3}6^{-1}$	$1^{2}2^{0}3^{4}6^{-2}$
$1^{2}2^{1}3^{-2}6^{3}$	$1^{2}2^{1}3^{-1}6^{2}$	$1^{2}2^{1}3^{0}6^{1}$	$1^{2}2^{1}3^{1}6^{0}$	$1^{2}2^{1}3^{2}6^{-1}$	$1^{2}2^{1}3^{3}6^{-2}$
$1^{2}2^{2}3^{-2}6^{2}$	$1^{2}2^{2}3^{-1}6^{1}$	$1^{2}2^{2}3^{1}6^{-1}$	$1^{2}2^{2}3^{2}6^{-2}$	$1^{2}2^{3}3^{-1}6^{0}$	$1^{2}2^{3}3^{0}6^{-1}$
$1^{3}2^{-2}3^{-2}6^{5}$	$1^{3}2^{-2}3^{-1}6^{4}$	$1^{3}2^{-2}3^{0}6^{3}$	$1^{3}2^{-2}3^{1}6^{2}$	$1^{3}2^{-2}3^{2}6^{1}$	$1^{3}2^{-2}3^{3}6^{0}$
$1^{3}2^{-1}3^{-3}6^{5}$	$1^{3}2^{-1}3^{-2}6^{4}$	$1^{3}2^{-1}3^{-1}6^{3}$	$1^{3}2^{-1}3^{0}6^{2}$	$1^{3}2^{-1}3^{1}6^{1}$	$1^{3}2^{-1}3^{2}6^{0}$
$1^{3}2^{-1}3^{3}6^{-1}$	$1^{3}2^{-1}3^{4}6^{-2}$	$1^{3}2^{0}3^{-3}6^{4}$	$1^{3}2^{0}3^{-2}6^{3}$	$1^{3}2^{0}3^{-1}6^{2}$	$1^{3}2^{0}3^{0}6^{1}$
$1^{3}2^{0}3^{2}6^{-1}$	$1^{3}2^{0}3^{3}6^{-2}$	$1^{3}2^{1}3^{-2}6^{2}$	$1^{3}2^{1}3^{-1}6^{1}$	$1^{3}2^{1}3^{1}6^{-1}$	$1^{3}2^{2}3^{-1}6^{0}$
$1^{3}2^{2}3^{0}6^{-1}$	$1^{4}2^{-3}3^{-2}6^{5}$	$1^{4}2^{-3}3^{-1}6^{4}$	$1^{4}2^{-3}3^{0}6^{3}$	$1^{4}2^{-2}3^{-3}6^{5}$	$1^{4}2^{-2}3^{-2}6^{4}$
$1^{4}2^{-2}3^{-1}6^{3}$	$1^{4}2^{-2}3^{0}6^{2}$	$1^{4}2^{-2}3^{1}6^{1}$	$1^{4}2^{-2}3^{2}6^{0}$	$1^{4}2^{-2}3^{3}6^{-1}$	$1^{4}2^{-1}3^{-3}6^{4}$
$1^{4}2^{-1}3^{-2}6^{3}$	$1^{4}2^{-1}3^{-1}6^{2}$	$1^{4}2^{-1}3^{0}6^{1}$	$1^{4}2^{-1}3^{1}6^{0}$	$1^{4}2^{0}3^{-2}6^{2}$	$1^{4}2^{0}3^{-1}6^{1}$
$1^{5}2^{-3}3^{-4}6^{6}$	$1^{5}2^{-3}3^{-3}6^{5}$	$1^{5}2^{-3}3^{-2}6^{4}$	$1^{5}2^{-3}3^{-1}6^{3}$	$1^{5}2^{-3}3^{0}6^{2}$	$1^{5}2^{-2}3^{-3}6^{4}$
$1^{5}2^{-2}3^{-2}6^{3}$	$1^{6}2^{-4}3^{-4}6^{6}$	$1^{6}2^{-4}3^{-3}6^{5}$
$N=6$ , $k=5/2$
$1^{-3}2^{4}3^{2}6^{2}$	$1^{-3}2^{4}3^{3}6^{1}$	$1^{-3}2^{4}3^{4}6^{0}$	$1^{-3}2^{5}3^{4}6^{-1}$	$1^{-3}2^{5}3^{5}6^{-2}$	$1^{-2}2^{2}3^{3}6^{2}$
$1^{-2}2^{3}3^{2}6^{2}$	$1^{-2}2^{3}3^{3}6^{1}$	$1^{-2}2^{3}3^{4}6^{0}$	$1^{-2}2^{4}3^{1}6^{2}$	$1^{-2}2^{4}3^{2}6^{1}$	$1^{-2}2^{4}3^{3}6^{0}$
$1^{-2}2^{4}3^{4}6^{-1}$	$1^{-2}2^{5}3^{5}6^{-3}$	$1^{-1}2^{1}3^{3}6^{2}$	$1^{-1}2^{2}3^{2}6^{2}$	$1^{-1}2^{2}3^{3}6^{1}$	$1^{-1}2^{3}3^{0}6^{3}$
$1^{-1}2^{3}3^{1}6^{2}$	$1^{-1}2^{3}3^{2}6^{1}$	$1^{-1}2^{3}3^{3}6^{0}$	$1^{-1}2^{3}3^{4}6^{-1}$	$1^{-1}2^{4}3^{2}6^{0}$	$1^{-1}2^{4}3^{3}6^{-1}$
$1^{-1}2^{4}3^{4}6^{-2}$	$1^{-1}2^{4}3^{5}6^{-3}$	$1^{0}2^{1}3^{1}6^{3}$	$1^{0}2^{1}3^{2}6^{2}$	$1^{0}2^{1}3^{3}6^{1}$	$1^{0}2^{2}3^{0}6^{3}$
$1^{0}2^{2}3^{1}6^{2}$	$1^{0}2^{2}3^{2}6^{1}$	$1^{0}2^{2}3^{3}6^{0}$	$1^{0}2^{2}3^{4}6^{-1}$	$1^{0}2^{3}3^{-1}6^{3}$	$1^{0}2^{3}3^{0}6^{2}$
$1^{0}2^{3}3^{1}6^{1}$	$1^{0}2^{3}3^{2}6^{0}$	$1^{0}2^{3}3^{3}6^{-1}$	$1^{0}2^{3}3^{4}6^{-2}$	$1^{0}2^{4}3^{3}6^{-2}$	$1^{0}2^{4}3^{4}6^{-3}$
$1^{1}2^{-1}3^{2}6^{3}$	$1^{1}2^{0}3^{1}6^{3}$	$1^{1}2^{0}3^{2}6^{2}$	$1^{1}2^{0}3^{3}6^{1}$	$1^{1}2^{1}3^{0}6^{3}$	$1^{1}2^{1}3^{1}6^{2}$
$1^{1}2^{1}3^{2}6^{1}$	$1^{1}2^{1}3^{3}6^{0}$	$1^{1}2^{2}3^{-2}6^{4}$	$1^{1}2^{2}3^{-1}6^{3}$	$1^{1}2^{2}3^{0}6^{2}$	$1^{1}2^{2}3^{1}6^{1}$
$1^{1}2^{2}3^{2}6^{0}$	$1^{1}2^{2}3^{3}6^{-1}$	$1^{1}2^{2}3^{4}6^{-2}$	$1^{1}2^{3}3^{0}6^{1}$	$1^{1}2^{3}3^{1}6^{0}$	$1^{1}2^{3}3^{2}6^{-1}$
$1^{1}2^{3}3^{3}6^{-2}$	$1^{1}2^{3}3^{4}6^{-3}$	$1^{2}2^{-2}3^{2}6^{3}$	$1^{2}2^{-1}3^{1}6^{3}$	$1^{2}2^{-1}3^{2}6^{2}$	$1^{2}2^{0}3^{-1}6^{4}$
$1^{2}2^{0}3^{0}6^{3}$	$1^{2}2^{0}3^{1}6^{2}$	$1^{2}2^{0}3^{2}6^{1}$	$1^{2}2^{1}3^{-2}6^{4}$	$1^{2}2^{1}3^{-1}6^{3}$	$1^{2}2^{1}3^{0}6^{2}$
$1^{2}2^{1}3^{1}6^{1}$	$1^{2}2^{1}3^{2}6^{0}$	$1^{2}2^{1}3^{3}6^{-1}$	$1^{2}2^{1}3^{4}6^{-2}$	$1^{2}2^{2}3^{-3}6^{4}$	$1^{2}2^{2}3^{-2}6^{3}$
$1^{2}2^{2}3^{-1}6^{2}$	$1^{2}2^{2}3^{0}6^{1}$	$1^{2}2^{2}3^{1}6^{0}$	$1^{2}2^{2}3^{2}6^{-1}$	$1^{2}2^{2}3^{3}6^{-2}$	$1^{2}2^{2}3^{4}6^{-3}$
$1^{2}2^{3}3^{1}6^{-1}$	$1^{2}2^{3}3^{2}6^{-2}$	$1^{3}2^{-2}3^{0}6^{4}$	$1^{3}2^{-2}3^{1}6^{3}$	$1^{3}2^{-2}3^{2}6^{2}$	$1^{3}2^{-1}3^{-1}6^{4}$
$1^{3}2^{-1}3^{0}6^{3}$	$1^{3}2^{-1}3^{1}6^{2}$	$1^{3}2^{-1}3^{2}6^{1}$	$1^{3}2^{-1}3^{3}6^{0}$	$1^{3}2^{0}3^{-2}6^{4}$	$1^{3}2^{0}3^{-1}6^{3}$
$1^{3}2^{0}3^{0}6^{2}$	$1^{3}2^{0}3^{1}6^{1}$	$1^{3}2^{0}3^{3}6^{-1}$	$1^{3}2^{1}3^{-3}6^{4}$	$1^{3}2^{1}3^{-2}6^{3}$	$1^{3}2^{1}3^{-1}6^{2}$
$1^{3}2^{1}3^{0}6^{1}$	$1^{3}2^{1}3^{1}6^{0}$	$1^{3}2^{2}3^{-2}6^{2}$	$1^{3}2^{2}3^{-1}6^{1}$	$1^{4}2^{-3}3^{0}6^{4}$	$1^{4}2^{-3}3^{1}6^{3}$
$1^{4}2^{-3}3^{2}6^{2}$	$1^{4}2^{-2}3^{-1}6^{4}$	$1^{4}2^{-2}3^{0}6^{3}$	$1^{4}2^{-2}3^{1}6^{2}$	$1^{4}2^{-2}3^{2}6^{1}$	$1^{4}2^{-1}3^{-3}6^{5}$
$1^{4}2^{-1}3^{-2}6^{4}$	$1^{4}2^{-1}3^{-1}6^{3}$	$1^{4}2^{-1}3^{0}6^{2}$	$1^{4}2^{0}3^{-3}6^{4}$	$1^{4}2^{0}3^{-2}6^{3}$	$1^{5}2^{-3}3^{-2}6^{5}$
$1^{5}2^{-3}3^{-1}6^{4}$	$1^{5}2^{-2}3^{-3}6^{5}$
$N=6$ , $k=3$
$1^{-2}2^{4}3^{3}6^{1}$	$1^{-1}2^{3}3^{2}6^{2}$	$1^{-1}2^{3}3^{3}6^{1}$	$1^{-1}2^{4}3^{4}6^{-1}$	$1^{0}2^{2}3^{2}6^{2}$	$1^{0}2^{2}3^{3}6^{1}$
$1^{0}2^{3}3^{1}6^{2}$	$1^{0}2^{3}3^{2}6^{1}$	$1^{0}2^{3}3^{3}6^{0}$	$1^{1}2^{1}3^{2}6^{2}$	$1^{1}2^{2}3^{0}6^{3}$	$1^{1}2^{2}3^{1}6^{2}$
$1^{1}2^{2}3^{2}6^{1}$	$1^{1}2^{2}3^{3}6^{0}$	$1^{1}2^{3}3^{2}6^{0}$	$1^{1}2^{3}3^{3}6^{-1}$	$1^{1}2^{3}3^{4}6^{-2}$	$1^{2}2^{0}3^{1}6^{3}$
$1^{2}2^{0}3^{2}6^{2}$	$1^{2}2^{1}3^{0}6^{3}$	$1^{2}2^{1}3^{1}6^{2}$	$1^{2}2^{1}3^{2}6^{1}$	$1^{2}2^{1}3^{3}6^{0}$	$1^{2}2^{2}3^{-1}6^{3}$
$1^{2}2^{2}3^{0}6^{2}$	$1^{2}2^{2}3^{1}6^{1}$	$1^{2}2^{2}3^{2}6^{0}$	$1^{2}2^{2}3^{3}6^{-1}$	$1^{3}2^{-1}3^{1}6^{3}$	$1^{3}2^{-1}3^{2}6^{2}$
$1^{3}2^{0}3^{0}6^{3}$	$1^{3}2^{0}3^{1}6^{2}$	$1^{3}2^{0}3^{2}6^{1}$	$1^{3}2^{1}3^{-2}6^{4}$	$1^{3}2^{1}3^{-1}6^{3}$	$1^{3}2^{1}3^{0}6^{2}$
$1^{4}2^{-2}3^{1}6^{3}$	$1^{4}2^{-1}3^{-1}6^{4}$
$N=6$ , $k=7/2$
$1^{1}2^{2}3^{2}6^{2}$	$1^{2}2^{1}3^{2}6^{2}$	$1^{2}2^{2}3^{1}6^{2}$	$1^{2}2^{2}3^{2}6^{1}$
$N=7$ , $k=1/2$
$1^{0}7^{1}$
$N=7$ , $k=1$
$1^{0}7^{2}$	$1^{1}7^{1}$
$N=7$ , $k=3/2$
$1^{0}7^{3}$	$1^{1}7^{2}$	$1^{2}7^{1}$
$N=7$ , $k=2$
$1^{1}7^{3}$	$1^{2}7^{2}$	$1^{3}7^{1}$
$N=7$ , $k=5/2$
$1^{2}7^{3}$	$1^{3}7^{2}$
$N=8$ , $k=1/2$
$1^{0}2^{-2}4^{5}8^{-2}$	$1^{0}2^{-1}4^{3}8^{-1}$	$1^{0}2^{0}4^{-1}8^{2}$	$1^{0}2^{0}4^{0}8^{1}$	$1^{0}2^{0}4^{2}8^{-1}$	$1^{0}2^{1}4^{-1}8^{1}$
$N=8$ , $k=1$
$1^{-2}2^{3}4^{3}8^{-2}$	$1^{-2}2^{4}4^{1}8^{-1}$	$1^{-2}2^{5}4^{-3}8^{2}$	$1^{-2}2^{5}4^{-2}8^{1}$	$1^{-2}2^{5}4^{0}8^{-1}$	$1^{-2}2^{6}4^{-3}8^{1}$
$1^{-1}2^{0}4^{5}8^{-2}$	$1^{-1}2^{1}4^{3}8^{-1}$	$1^{-1}2^{1}4^{4}8^{-2}$	$1^{-1}2^{2}4^{-1}8^{2}$	$1^{-1}2^{2}4^{0}8^{1}$	$1^{-1}2^{2}4^{2}8^{-1}$
$1^{-1}2^{3}4^{-2}8^{2}$	$1^{-1}2^{3}4^{-1}8^{1}$	$1^{-1}2^{3}4^{1}8^{-1}$	$1^{-1}2^{4}4^{-2}8^{1}$	$1^{0}2^{-3}4^{7}8^{-2}$	$1^{0}2^{-3}4^{8}8^{-3}$
$1^{0}2^{-2}4^{5}8^{-1}$	$1^{0}2^{-2}4^{6}8^{-2}$	$1^{0}2^{-2}4^{7}8^{-3}$	$1^{0}2^{-1}4^{1}8^{2}$	$1^{0}2^{-1}4^{2}8^{1}$	$1^{0}2^{-1}4^{4}8^{-1}$
$1^{0}2^{-1}4^{5}8^{-2}$	$1^{0}2^{0}4^{-1}8^{3}$	$1^{0}2^{0}4^{0}8^{2}$	$1^{0}2^{0}4^{1}8^{1}$	$1^{0}2^{0}4^{3}8^{-1}$	$1^{0}2^{0}4^{4}8^{-2}$
$1^{0}2^{1}4^{-2}8^{3}$	$1^{0}2^{1}4^{-1}8^{2}$	$1^{0}2^{1}4^{0}8^{1}$	$1^{0}2^{1}4^{2}8^{-1}$	$1^{0}2^{2}4^{-2}8^{2}$	$1^{0}2^{2}4^{-1}8^{1}$
$1^{0}2^{2}4^{1}8^{-1}$	$1^{0}2^{3}4^{-2}8^{1}$	$1^{1}2^{-3}4^{6}8^{-2}$	$1^{1}2^{-2}4^{4}8^{-1}$	$1^{1}2^{-2}4^{5}8^{-2}$	$1^{1}2^{-1}4^{0}8^{2}$
$1^{1}2^{-1}4^{1}8^{1}$	$1^{1}2^{-1}4^{3}8^{-1}$	$1^{1}2^{0}4^{-1}8^{2}$	$1^{1}2^{0}4^{0}8^{1}$	$1^{1}2^{0}4^{2}8^{-1}$	$1^{1}2^{1}4^{-1}8^{1}$
$1^{2}2^{-3}4^{5}8^{-2}$	$1^{2}2^{-2}4^{3}8^{-1}$	$1^{2}2^{-1}4^{-1}8^{2}$	$1^{2}2^{-1}4^{0}8^{1}$	$1^{2}2^{-1}4^{2}8^{-1}$	$1^{2}2^{0}4^{-1}8^{1}$
$N=8$ , $k=3/2$
$1^{-3}2^{5}4^{3}8^{-2}$	$1^{-3}2^{6}4^{1}8^{-1}$	$1^{-3}2^{6}4^{2}8^{-2}$	$1^{-3}2^{7}4^{-3}8^{2}$	$1^{-3}2^{7}4^{-2}8^{1}$	$1^{-3}2^{7}4^{0}8^{-1}$
$1^{-3}2^{8}4^{-4}8^{2}$	$1^{-3}2^{8}4^{-3}8^{1}$	$1^{-3}2^{8}4^{-1}8^{-1}$	$1^{-3}2^{9}4^{-4}8^{1}$	$1^{-2}2^{2}4^{5}8^{-2}$	$1^{-2}2^{2}4^{6}8^{-3}$
$1^{-2}2^{3}4^{3}8^{-1}$	$1^{-2}2^{3}4^{4}8^{-2}$	$1^{-2}2^{3}4^{5}8^{-3}$	$1^{-2}2^{4}4^{-1}8^{2}$	$1^{-2}2^{4}4^{0}8^{1}$	$1^{-2}2^{4}4^{2}8^{-1}$
$1^{-2}2^{4}4^{3}8^{-2}$	$1^{-2}2^{5}4^{-3}8^{3}$	$1^{-2}2^{5}4^{-2}8^{2}$	$1^{-2}2^{5}4^{-1}8^{1}$	$1^{-2}2^{5}4^{1}8^{-1}$	$1^{-2}2^{5}4^{2}8^{-2}$
$1^{-2}2^{6}4^{-4}8^{3}$	$1^{-2}2^{6}4^{-3}8^{2}$	$1^{-2}2^{6}4^{-2}8^{1}$	$1^{-2}2^{6}4^{0}8^{-1}$	$1^{-2}2^{7}4^{-4}8^{2}$	$1^{-2}2^{7}4^{-3}8^{1}$
$1^{-2}2^{7}4^{-1}8^{-1}$	$1^{-2}2^{8}4^{-4}8^{1}$	$1^{-1}2^{-1}4^{7}8^{-2}$	$1^{-1}2^{-1}4^{8}8^{-3}$	$1^{-1}2^{0}4^{5}8^{-1}$	$1^{-1}2^{0}4^{6}8^{-2}$
$1^{-1}2^{0}4^{7}8^{-3}$	$1^{-1}2^{1}4^{1}8^{2}$	$1^{-1}2^{1}4^{2}8^{1}$	$1^{-1}2^{1}4^{4}8^{-1}$	$1^{-1}2^{1}4^{5}8^{-2}$	$1^{-1}2^{1}4^{6}8^{-3}$
$1^{-1}2^{2}4^{-1}8^{3}$	$1^{-1}2^{2}4^{0}8^{2}$	$1^{-1}2^{2}4^{1}8^{1}$	$1^{-1}2^{2}4^{3}8^{-1}$	$1^{-1}2^{2}4^{4}8^{-2}$	$1^{-1}2^{3}4^{-2}8^{3}$
$1^{-1}2^{3}4^{-1}8^{2}$	$1^{-1}2^{3}4^{0}8^{1}$	$1^{-1}2^{3}4^{2}8^{-1}$	$1^{-1}2^{3}4^{3}8^{-2}$	$1^{-1}2^{4}4^{-3}8^{3}$	$1^{-1}2^{4}4^{-2}8^{2}$
$1^{-1}2^{4}4^{-1}8^{1}$	$1^{-1}2^{4}4^{1}8^{-1}$	$1^{-1}2^{5}4^{-3}8^{2}$	$1^{-1}2^{5}4^{-2}8^{1}$	$1^{-1}2^{5}4^{0}8^{-1}$	$1^{-1}2^{6}4^{-3}8^{1}$
$1^{0}2^{-3}4^{7}8^{-1}$	$1^{0}2^{-3}4^{8}8^{-2}$	$1^{0}2^{-2}4^{4}8^{1}$	$1^{0}2^{-2}4^{6}8^{-1}$	$1^{0}2^{-2}4^{7}8^{-2}$	$1^{0}2^{-2}4^{8}8^{-3}$
$1^{0}2^{-1}4^{2}8^{2}$	$1^{0}2^{-1}4^{3}8^{1}$	$1^{0}2^{-1}4^{5}8^{-1}$	$1^{0}2^{-1}4^{6}8^{-2}$	$1^{0}2^{-1}4^{7}8^{-3}$	$1^{0}2^{0}4^{1}8^{2}$
$1^{0}2^{0}4^{2}8^{1}$	$1^{0}2^{0}4^{4}8^{-1}$	$1^{0}2^{0}4^{5}8^{-2}$	$1^{0}2^{0}4^{6}8^{-3}$	$1^{0}2^{1}4^{-1}8^{3}$	$1^{0}2^{1}4^{0}8^{2}$
$1^{0}2^{1}4^{1}8^{1}$	$1^{0}2^{1}4^{3}8^{-1}$	$1^{0}2^{1}4^{4}8^{-2}$	$1^{0}2^{2}4^{-2}8^{3}$	$1^{0}2^{2}4^{-1}8^{2}$	$1^{0}2^{2}4^{0}8^{1}$
$1^{0}2^{2}4^{2}8^{-1}$	$1^{0}2^{2}4^{3}8^{-2}$	$1^{0}2^{3}4^{-3}8^{3}$	$1^{0}2^{3}4^{-2}8^{2}$	$1^{0}2^{3}4^{-1}8^{1}$	$1^{0}2^{3}4^{1}8^{-1}$
$1^{0}2^{4}4^{-3}8^{2}$	$1^{0}2^{4}4^{-2}8^{1}$	$1^{0}2^{4}4^{0}8^{-1}$	$1^{0}2^{5}4^{-3}8^{1}$	$1^{1}2^{-4}4^{8}8^{-2}$	$1^{1}2^{-4}4^{9}8^{-3}$
$1^{1}2^{-3}4^{6}8^{-1}$	$1^{1}2^{-3}4^{7}8^{-2}$	$1^{1}2^{-3}4^{8}8^{-3}$	$1^{1}2^{-2}4^{2}8^{2}$	$1^{1}2^{-2}4^{3}8^{1}$	$1^{1}2^{-2}4^{5}8^{-1}$
$1^{1}2^{-2}4^{6}8^{-2}$	$1^{1}2^{-2}4^{7}8^{-3}$	$1^{1}2^{-1}4^{0}8^{3}$	$1^{1}2^{-1}4^{1}8^{2}$	$1^{1}2^{-1}4^{2}8^{1}$	$1^{1}2^{-1}4^{4}8^{-1}$
$1^{1}2^{-1}4^{5}8^{-2}$	$1^{1}2^{0}4^{-1}8^{3}$	$1^{1}2^{0}4^{0}8^{2}$	$1^{1}2^{0}4^{1}8^{1}$	$1^{1}2^{0}4^{3}8^{-1}$	$1^{1}2^{0}4^{4}8^{-2}$
$1^{1}2^{1}4^{-2}8^{3}$	$1^{1}2^{1}4^{-1}8^{2}$	$1^{1}2^{1}4^{0}8^{1}$	$1^{1}2^{1}4^{2}8^{-1}$	$1^{1}2^{2}4^{-2}8^{2}$	$1^{1}2^{2}4^{-1}8^{1}$
$1^{1}2^{2}4^{1}8^{-1}$	$1^{1}2^{3}4^{-2}8^{1}$	$1^{2}2^{-4}4^{7}8^{-2}$	$1^{2}2^{-4}4^{8}8^{-3}$	$1^{2}2^{-3}4^{5}8^{-1}$	$1^{2}2^{-3}4^{6}8^{-2}$
$1^{2}2^{-3}4^{7}8^{-3}$	$1^{2}2^{-2}4^{1}8^{2}$	$1^{2}2^{-2}4^{2}8^{1}$	$1^{2}2^{-2}4^{4}8^{-1}$	$1^{2}2^{-2}4^{5}8^{-2}$	$1^{2}2^{-1}4^{-1}8^{3}$
$1^{2}2^{-1}4^{0}8^{2}$	$1^{2}2^{-1}4^{1}8^{1}$	$1^{2}2^{-1}4^{3}8^{-1}$	$1^{2}2^{-1}4^{4}8^{-2}$	$1^{2}2^{0}4^{-2}8^{3}$	$1^{2}2^{0}4^{-1}8^{2}$
$1^{2}2^{0}4^{0}8^{1}$	$1^{2}2^{0}4^{2}8^{-1}$	$1^{2}2^{1}4^{-2}8^{2}$	$1^{2}2^{1}4^{-1}8^{1}$	$1^{2}2^{1}4^{1}8^{-1}$	$1^{2}2^{2}4^{-2}8^{1}$
$1^{3}2^{-4}4^{6}8^{-2}$	$1^{3}2^{-3}4^{4}8^{-1}$	$1^{3}2^{-3}4^{5}8^{-2}$	$1^{3}2^{-2}4^{0}8^{2}$	$1^{3}2^{-2}4^{1}8^{1}$	$1^{3}2^{-2}4^{3}8^{-1}$
$1^{3}2^{-1}4^{-1}8^{2}$	$1^{3}2^{-1}4^{0}8^{1}$	$1^{3}2^{-1}4^{2}8^{-1}$	$1^{3}2^{0}4^{-1}8^{1}$
$N=8$ , $k=2$
$1^{-3}2^{4}4^{5}8^{-2}$	$1^{-3}2^{4}4^{6}8^{-3}$	$1^{-3}2^{5}4^{3}8^{-1}$	$1^{-3}2^{5}4^{4}8^{-2}$	$1^{-3}2^{5}4^{5}8^{-3}$	$1^{-3}2^{6}4^{-1}8^{2}$
$1^{-3}2^{6}4^{0}8^{1}$	$1^{-3}2^{6}4^{2}8^{-1}$	$1^{-3}2^{6}4^{3}8^{-2}$	$1^{-3}2^{6}4^{4}8^{-3}$	$1^{-3}2^{7}4^{-3}8^{3}$	$1^{-3}2^{7}4^{-2}8^{2}$
$1^{-3}2^{7}4^{-1}8^{1}$	$1^{-3}2^{7}4^{1}8^{-1}$	$1^{-3}2^{8}4^{-4}8^{3}$	$1^{-3}2^{8}4^{-3}8^{2}$	$1^{-3}2^{8}4^{-2}8^{1}$	$1^{-3}2^{9}4^{-5}8^{3}$
$1^{-2}2^{2}4^{5}8^{-1}$	$1^{-2}2^{2}4^{6}8^{-2}$	$1^{-2}2^{3}4^{2}8^{1}$	$1^{-2}2^{3}4^{4}8^{-1}$	$1^{-2}2^{3}4^{5}8^{-2}$	$1^{-2}2^{3}4^{6}8^{-3}$
$1^{-2}2^{4}4^{0}8^{2}$	$1^{-2}2^{4}4^{1}8^{1}$	$1^{-2}2^{4}4^{3}8^{-1}$	$1^{-2}2^{4}4^{4}8^{-2}$	$1^{-2}2^{4}4^{5}8^{-3}$	$1^{-2}2^{5}4^{-1}8^{2}$
$1^{-2}2^{5}4^{0}8^{1}$	$1^{-2}2^{5}4^{2}8^{-1}$	$1^{-2}2^{5}4^{3}8^{-2}$	$1^{-2}2^{5}4^{4}8^{-3}$	$1^{-2}2^{6}4^{-3}8^{3}$	$1^{-2}2^{6}4^{-2}8^{2}$
$1^{-2}2^{6}4^{-1}8^{1}$	$1^{-2}2^{6}4^{1}8^{-1}$	$1^{-2}2^{6}4^{2}8^{-2}$	$1^{-2}2^{7}4^{-4}8^{3}$	$1^{-2}2^{7}4^{-3}8^{2}$	$1^{-2}2^{7}4^{-2}8^{1}$
$1^{-2}2^{7}4^{0}8^{-1}$	$1^{-2}2^{8}4^{-5}8^{3}$	$1^{-2}2^{8}4^{-4}8^{2}$	$1^{-2}2^{8}4^{-3}8^{1}$	$1^{-1}2^{-1}4^{7}8^{-1}$	$1^{-1}2^{0}4^{4}8^{1}$
$1^{-1}2^{0}4^{6}8^{-1}$	$1^{-1}2^{0}4^{7}8^{-2}$	$1^{-1}2^{1}4^{3}8^{1}$	$1^{-1}2^{1}4^{5}8^{-1}$	$1^{-1}2^{1}4^{6}8^{-2}$	$1^{-1}2^{1}4^{7}8^{-3}$
$1^{-1}2^{2}4^{1}8^{2}$	$1^{-1}2^{2}4^{2}8^{1}$	$1^{-1}2^{2}4^{4}8^{-1}$	$1^{-1}2^{2}4^{5}8^{-2}$	$1^{-1}2^{2}4^{6}8^{-3}$	$1^{-1}2^{3}4^{0}8^{2}$
$1^{-1}2^{3}4^{1}8^{1}$	$1^{-1}2^{3}4^{3}8^{-1}$	$1^{-1}2^{3}4^{4}8^{-2}$	$1^{-1}2^{3}4^{5}8^{-3}$	$1^{-1}2^{4}4^{-2}8^{3}$	$1^{-1}2^{4}4^{-1}8^{2}$
$1^{-1}2^{4}4^{0}8^{1}$	$1^{-1}2^{4}4^{2}8^{-1}$	$1^{-1}2^{4}4^{3}8^{-2}$	$1^{-1}2^{5}4^{-3}8^{3}$	$1^{-1}2^{5}4^{-2}8^{2}$	$1^{-1}2^{5}4^{-1}8^{1}$
$1^{-1}2^{5}4^{1}8^{-1}$	$1^{-1}2^{5}4^{2}8^{-2}$	$1^{-1}2^{6}4^{-4}8^{3}$	$1^{-1}2^{6}4^{-3}8^{2}$	$1^{-1}2^{6}4^{-2}8^{1}$	$1^{-1}2^{6}4^{0}8^{-1}$
$1^{-1}2^{7}4^{-4}8^{2}$	$1^{-1}2^{7}4^{-3}8^{1}$	$1^{-1}2^{7}4^{-1}8^{-1}$	$1^{-1}2^{8}4^{-4}8^{1}$	$1^{0}2^{-2}4^{7}8^{-1}$	$1^{0}2^{-1}4^{4}8^{1}$
$1^{0}2^{-1}4^{6}8^{-1}$	$1^{0}2^{-1}4^{7}8^{-2}$	$1^{0}2^{0}4^{3}8^{1}$	$1^{0}2^{0}4^{5}8^{-1}$	$1^{0}2^{0}4^{6}8^{-2}$	$1^{0}2^{0}4^{7}8^{-3}$
$1^{0}2^{1}4^{1}8^{2}$	$1^{0}2^{1}4^{2}8^{1}$	$1^{0}2^{1}4^{4}8^{-1}$	$1^{0}2^{1}4^{5}8^{-2}$	$1^{0}2^{1}4^{6}8^{-3}$	$1^{0}2^{2}4^{0}8^{2}$
$1^{0}2^{2}4^{1}8^{1}$	$1^{0}2^{2}4^{3}8^{-1}$	$1^{0}2^{2}4^{4}8^{-2}$	$1^{0}2^{2}4^{5}8^{-3}$	$1^{0}2^{3}4^{-2}8^{3}$	$1^{0}2^{3}4^{-1}8^{2}$
$1^{0}2^{3}4^{0}8^{1}$	$1^{0}2^{3}4^{2}8^{-1}$	$1^{0}2^{3}4^{3}8^{-2}$	$1^{0}2^{4}4^{-3}8^{3}$	$1^{0}2^{4}4^{-2}8^{2}$	$1^{0}2^{4}4^{-1}8^{1}$
$1^{0}2^{4}4^{1}8^{-1}$	$1^{0}2^{4}4^{2}8^{-2}$	$1^{0}2^{5}4^{-4}8^{3}$	$1^{0}2^{5}4^{-3}8^{2}$	$1^{0}2^{5}4^{-2}8^{1}$	$1^{0}2^{5}4^{0}8^{-1}$
$1^{0}2^{6}4^{-4}8^{2}$	$1^{0}2^{6}4^{-3}8^{1}$	$1^{0}2^{6}4^{-1}8^{-1}$	$1^{0}2^{7}4^{-4}8^{1}$	$1^{1}2^{-4}4^{8}8^{-1}$	$1^{1}2^{-3}4^{5}8^{1}$
$1^{1}2^{-3}4^{7}8^{-1}$	$1^{1}2^{-3}4^{8}8^{-2}$	$1^{1}2^{-2}4^{4}8^{1}$	$1^{1}2^{-2}4^{6}8^{-1}$	$1^{1}2^{-2}4^{7}8^{-2}$	$1^{1}2^{-2}4^{8}8^{-3}$
$1^{1}2^{-1}4^{2}8^{2}$	$1^{1}2^{-1}4^{3}8^{1}$	$1^{1}2^{-1}4^{5}8^{-1}$	$1^{1}2^{-1}4^{6}8^{-2}$	$1^{1}2^{-1}4^{7}8^{-3}$	$1^{1}2^{0}4^{1}8^{2}$
$1^{1}2^{0}4^{2}8^{1}$	$1^{1}2^{0}4^{4}8^{-1}$	$1^{1}2^{0}4^{5}8^{-2}$	$1^{1}2^{0}4^{6}8^{-3}$	$1^{1}2^{1}4^{-1}8^{3}$	$1^{1}2^{1}4^{0}8^{2}$
$1^{1}2^{1}4^{1}8^{1}$	$1^{1}2^{1}4^{3}8^{-1}$	$1^{1}2^{1}4^{4}8^{-2}$	$1^{1}2^{2}4^{-2}8^{3}$	$1^{1}2^{2}4^{-1}8^{2}$	$1^{1}2^{2}4^{0}8^{1}$
$1^{1}2^{2}4^{2}8^{-1}$	$1^{1}2^{2}4^{3}8^{-2}$	$1^{1}2^{3}4^{-3}8^{3}$	$1^{1}2^{3}4^{-2}8^{2}$	$1^{1}2^{3}4^{-1}8^{1}$	$1^{1}2^{3}4^{1}8^{-1}$
$1^{1}2^{4}4^{-3}8^{2}$	$1^{1}2^{4}4^{-2}8^{1}$	$1^{1}2^{4}4^{0}8^{-1}$	$1^{1}2^{5}4^{-3}8^{1}$	$1^{2}2^{-4}4^{7}8^{-1}$	$1^{2}2^{-4}4^{8}8^{-2}$
$1^{2}2^{-3}4^{4}8^{1}$	$1^{2}2^{-3}4^{6}8^{-1}$	$1^{2}2^{-3}4^{7}8^{-2}$	$1^{2}2^{-3}4^{8}8^{-3}$	$1^{2}2^{-2}4^{2}8^{2}$	$1^{2}2^{-2}4^{3}8^{1}$
$1^{2}2^{-2}4^{5}8^{-1}$	$1^{2}2^{-2}4^{6}8^{-2}$	$1^{2}2^{-2}4^{7}8^{-3}$	$1^{2}2^{-1}4^{1}8^{2}$	$1^{2}2^{-1}4^{2}8^{1}$	$1^{2}2^{-1}4^{4}8^{-1}$
$1^{2}2^{-1}4^{5}8^{-2}$	$1^{2}2^{-1}4^{6}8^{-3}$	$1^{2}2^{0}4^{-1}8^{3}$	$1^{2}2^{0}4^{0}8^{2}$	$1^{2}2^{0}4^{1}8^{1}$	$1^{2}2^{0}4^{3}8^{-1}$
$1^{2}2^{0}4^{4}8^{-2}$	$1^{2}2^{1}4^{-2}8^{3}$	$1^{2}2^{1}4^{-1}8^{2}$	$1^{2}2^{1}4^{0}8^{1}$	$1^{2}2^{1}4^{2}8^{-1}$	$1^{2}2^{2}4^{-3}8^{3}$
$1^{2}2^{2}4^{-2}8^{2}$	$1^{2}2^{2}4^{-1}8^{1}$	$1^{3}2^{-5}4^{8}8^{-2}$	$1^{3}2^{-5}4^{9}8^{-3}$	$1^{3}2^{-4}4^{6}8^{-1}$	$1^{3}2^{-4}4^{7}8^{-2}$
$1^{3}2^{-4}4^{8}8^{-3}$	$1^{3}2^{-3}4^{2}8^{2}$	$1^{3}2^{-3}4^{3}8^{1}$	$1^{3}2^{-3}4^{5}8^{-1}$	$1^{3}2^{-3}4^{6}8^{-2}$	$1^{3}2^{-3}4^{7}8^{-3}$
$1^{3}2^{-2}4^{0}8^{3}$	$1^{3}2^{-2}4^{1}8^{2}$	$1^{3}2^{-2}4^{2}8^{1}$	$1^{3}2^{-2}4^{4}8^{-1}$	$1^{3}2^{-1}4^{-1}8^{3}$	$1^{3}2^{-1}4^{0}8^{2}$
$1^{3}2^{-1}4^{1}8^{1}$	$1^{3}2^{0}4^{-2}8^{3}$
$N=8$ , $k=5/2$
$1^{-3}2^{4}4^{5}8^{-1}$	$1^{-3}2^{5}4^{2}8^{1}$	$1^{-3}2^{5}4^{4}8^{-1}$	$1^{-3}2^{5}4^{5}8^{-2}$	$1^{-3}2^{6}4^{1}8^{1}$	$1^{-3}2^{6}4^{3}8^{-1}$
$1^{-3}2^{7}4^{-1}8^{2}$	$1^{-3}2^{7}4^{0}8^{1}$	$1^{-2}2^{3}4^{5}8^{-1}$	$1^{-2}2^{4}4^{2}8^{1}$	$1^{-2}2^{4}4^{4}8^{-1}$	$1^{-2}2^{4}4^{5}8^{-2}$
$1^{-2}2^{5}4^{1}8^{1}$	$1^{-2}2^{5}4^{3}8^{-1}$	$1^{-2}2^{5}4^{4}8^{-2}$	$1^{-2}2^{5}4^{5}8^{-3}$	$1^{-2}2^{6}4^{-1}8^{2}$	$1^{-2}2^{6}4^{0}8^{1}$
$1^{-2}2^{6}4^{2}8^{-1}$	$1^{-2}2^{7}4^{-2}8^{2}$	$1^{-2}2^{7}4^{-1}8^{1}$	$1^{-2}2^{8}4^{-4}8^{3}$	$1^{-1}2^{1}4^{6}8^{-1}$	$1^{-1}2^{2}4^{3}8^{1}$
$1^{-1}2^{2}4^{5}8^{-1}$	$1^{-1}2^{2}4^{6}8^{-2}$	$1^{-1}2^{3}4^{2}8^{1}$	$1^{-1}2^{3}4^{4}8^{-1}$	$1^{-1}2^{3}4^{5}8^{-2}$	$1^{-1}2^{3}4^{6}8^{-3}$
$1^{-1}2^{4}4^{0}8^{2}$	$1^{-1}2^{4}4^{1}8^{1}$	$1^{-1}2^{4}4^{3}8^{-1}$	$1^{-1}2^{4}4^{4}8^{-2}$	$1^{-1}2^{4}4^{5}8^{-3}$	$1^{-1}2^{5}4^{-1}8^{2}$
$1^{-1}2^{5}4^{0}8^{1}$	$1^{-1}2^{5}4^{2}8^{-1}$	$1^{-1}2^{5}4^{3}8^{-2}$	$1^{-1}2^{5}4^{4}8^{-3}$	$1^{-1}2^{6}4^{-3}8^{3}$	$1^{-1}2^{6}4^{-2}8^{2}$
$1^{-1}2^{6}4^{-1}8^{1}$	$1^{-1}2^{6}4^{1}8^{-1}$	$1^{-1}2^{7}4^{-4}8^{3}$	$1^{-1}2^{7}4^{-3}8^{2}$	$1^{-1}2^{7}4^{-2}8^{1}$	$1^{-1}2^{8}4^{-5}8^{3}$
$1^{0}2^{0}4^{6}8^{-1}$	$1^{0}2^{1}4^{3}8^{1}$	$1^{0}2^{1}4^{5}8^{-1}$	$1^{0}2^{1}4^{6}8^{-2}$	$1^{0}2^{2}4^{2}8^{1}$	$1^{0}2^{2}4^{4}8^{-1}$
$1^{0}2^{2}4^{5}8^{-2}$	$1^{0}2^{2}4^{6}8^{-3}$	$1^{0}2^{3}4^{0}8^{2}$	$1^{0}2^{3}4^{1}8^{1}$	$1^{0}2^{3}4^{3}8^{-1}$	$1^{0}2^{3}4^{4}8^{-2}$
$1^{0}2^{3}4^{5}8^{-3}$	$1^{0}2^{4}4^{-1}8^{2}$	$1^{0}2^{4}4^{0}8^{1}$	$1^{0}2^{4}4^{2}8^{-1}$	$1^{0}2^{4}4^{3}8^{-2}$	$1^{0}2^{4}4^{4}8^{-3}$
$1^{0}2^{5}4^{-3}8^{3}$	$1^{0}2^{5}4^{-2}8^{2}$	$1^{0}2^{5}4^{-1}8^{1}$	$1^{0}2^{5}4^{1}8^{-1}$	$1^{0}2^{5}4^{2}8^{-2}$	$1^{0}2^{6}4^{-4}8^{3}$
$1^{0}2^{6}4^{-3}8^{2}$	$1^{0}2^{6}4^{-2}8^{1}$	$1^{0}2^{6}4^{0}8^{-1}$	$1^{0}2^{7}4^{-5}8^{3}$	$1^{0}2^{7}4^{-4}8^{2}$	$1^{0}2^{7}4^{-3}8^{1}$
$1^{1}2^{-2}4^{7}8^{-1}$	$1^{1}2^{-1}4^{4}8^{1}$	$1^{1}2^{-1}4^{6}8^{-1}$	$1^{1}2^{-1}4^{7}8^{-2}$	$1^{1}2^{0}4^{3}8^{1}$	$1^{1}2^{0}4^{5}8^{-1}$
$1^{1}2^{0}4^{6}8^{-2}$	$1^{1}2^{0}4^{7}8^{-3}$	$1^{1}2^{1}4^{1}8^{2}$	$1^{1}2^{1}4^{2}8^{1}$	$1^{1}2^{1}4^{4}8^{-1}$	$1^{1}2^{1}4^{5}8^{-2}$
$1^{1}2^{1}4^{6}8^{-3}$	$1^{1}2^{2}4^{0}8^{2}$	$1^{1}2^{2}4^{1}8^{1}$	$1^{1}2^{2}4^{3}8^{-1}$	$1^{1}2^{2}4^{4}8^{-2}$	$1^{1}2^{2}4^{5}8^{-3}$
$1^{1}2^{3}4^{-2}8^{3}$	$1^{1}2^{3}4^{-1}8^{2}$	$1^{1}2^{3}4^{0}8^{1}$	$1^{1}2^{3}4^{2}8^{-1}$	$1^{1}2^{4}4^{-3}8^{3}$	$1^{1}2^{4}4^{-2}8^{2}$
$1^{1}2^{4}4^{-1}8^{1}$	$1^{1}2^{5}4^{-4}8^{3}$	$1^{2}2^{-3}4^{7}8^{-1}$	$1^{2}2^{-2}4^{4}8^{1}$	$1^{2}2^{-2}4^{6}8^{-1}$	$1^{2}2^{-2}4^{7}8^{-2}$
$1^{2}2^{-1}4^{3}8^{1}$	$1^{2}2^{-1}4^{5}8^{-1}$	$1^{2}2^{-1}4^{6}8^{-2}$	$1^{2}2^{-1}4^{7}8^{-3}$	$1^{2}2^{0}4^{1}8^{2}$	$1^{2}2^{0}4^{2}8^{1}$
$1^{2}2^{0}4^{4}8^{-1}$	$1^{2}2^{1}4^{0}8^{2}$	$1^{2}2^{1}4^{1}8^{1}$	$1^{2}2^{2}4^{-2}8^{3}$	$1^{3}2^{-5}4^{8}8^{-1}$	$1^{3}2^{-4}4^{5}8^{1}$
$1^{3}2^{-4}4^{7}8^{-1}$	$1^{3}2^{-4}4^{8}8^{-2}$	$1^{3}2^{-3}4^{4}8^{1}$	$1^{3}2^{-3}4^{6}8^{-1}$	$1^{3}2^{-2}4^{2}8^{2}$	$1^{3}2^{-2}4^{3}8^{1}$
$N=8$ , $k=3$
$1^{-2}2^{5}4^{4}8^{-1}$	$1^{-2}2^{6}4^{1}8^{1}$	$1^{-1}2^{3}4^{5}8^{-1}$	$1^{-1}2^{4}4^{2}8^{1}$	$1^{-1}2^{4}4^{4}8^{-1}$	$1^{-1}2^{4}4^{5}8^{-2}$
$1^{-1}2^{5}4^{1}8^{1}$	$1^{-1}2^{5}4^{3}8^{-1}$	$1^{-1}2^{6}4^{-1}8^{2}$	$1^{-1}2^{6}4^{0}8^{1}$	$1^{0}2^{2}4^{5}8^{-1}$	$1^{0}2^{3}4^{2}8^{1}$
$1^{0}2^{3}4^{4}8^{-1}$	$1^{0}2^{3}4^{5}8^{-2}$	$1^{0}2^{4}4^{1}8^{1}$	$1^{0}2^{4}4^{3}8^{-1}$	$1^{0}2^{4}4^{4}8^{-2}$	$1^{0}2^{4}4^{5}8^{-3}$
$1^{0}2^{5}4^{-1}8^{2}$	$1^{0}2^{5}4^{0}8^{1}$	$1^{0}2^{5}4^{2}8^{-1}$	$1^{0}2^{6}4^{-2}8^{2}$	$1^{0}2^{6}4^{-1}8^{1}$	$1^{0}2^{7}4^{-4}8^{3}$
$1^{1}2^{0}4^{6}8^{-1}$	$1^{1}2^{1}4^{3}8^{1}$	$1^{1}2^{1}4^{5}8^{-1}$	$1^{1}2^{1}4^{6}8^{-2}$	$1^{1}2^{2}4^{2}8^{1}$	$1^{1}2^{2}4^{4}8^{-1}$
$1^{1}2^{3}4^{0}8^{2}$	$1^{1}2^{3}4^{1}8^{1}$	$1^{2}2^{-1}4^{6}8^{-1}$	$1^{2}2^{0}4^{3}8^{1}$
$N=8$ , $k=7/2$
$1^{0}2^{4}4^{4}8^{-1}$	$1^{0}2^{5}4^{1}8^{1}$
$N=9$ , $k=1/2$
$1^{0}3^{0}9^{1}$
$N=9$ , $k=1$
$1^{-1}3^{4}9^{-1}$	$1^{0}3^{0}9^{2}$	$1^{0}3^{1}9^{1}$	$1^{0}3^{3}9^{-1}$	$1^{1}3^{-1}9^{2}$	$1^{1}3^{0}9^{1}$
$1^{2}3^{-1}9^{1}$
$N=9$ , $k=3/2$
$1^{-2}3^{7}9^{-2}$	$1^{-1}3^{3}9^{1}$	$1^{-1}3^{5}9^{-1}$	$1^{0}3^{1}9^{2}$	$1^{0}3^{2}9^{1}$	$1^{0}3^{4}9^{-1}$
$1^{1}3^{0}9^{2}$	$1^{1}3^{1}9^{1}$	$1^{1}3^{3}9^{-1}$	$1^{2}3^{-1}9^{2}$	$1^{2}3^{0}9^{1}$
$N=9$ , $k=2$
$1^{-2}3^{7}9^{-1}$	$1^{-2}3^{8}9^{-2}$	$1^{-1}3^{4}9^{1}$	$1^{-1}3^{6}9^{-1}$	$1^{-1}3^{7}9^{-2}$	$1^{0}3^{3}9^{1}$
$1^{0}3^{5}9^{-1}$	$1^{0}3^{6}9^{-2}$	$1^{1}3^{1}9^{2}$	$1^{1}3^{2}9^{1}$	$1^{1}3^{4}9^{-1}$	$1^{2}3^{0}9^{2}$
$1^{2}3^{1}9^{1}$
$N=9$ , $k=5/2$
$1^{-2}3^{8}9^{-1}$	$1^{-2}3^{9}9^{-2}$	$1^{-1}3^{5}9^{1}$	$1^{-1}3^{7}9^{-1}$	$1^{-1}3^{8}9^{-2}$	$1^{0}3^{4}9^{1}$
$1^{0}3^{6}9^{-1}$	$1^{0}3^{7}9^{-2}$	$1^{1}3^{3}9^{1}$	$1^{1}3^{5}9^{-1}$	$1^{2}3^{1}9^{2}$
$N=9$ , $k=3$
$1^{-1}3^{8}9^{-1}$	$1^{0}3^{7}9^{-1}$	$1^{0}3^{8}9^{-2}$	$1^{1}3^{4}9^{1}$
$N=9$ , $k=7/2$
$1^{0}3^{8}9^{-1}$
$N=10$ , $k=1/2$
$1^{0}2^{0}5^{-1}10^{2}$	$1^{0}2^{0}5^{0}10^{1}$	$1^{0}2^{0}5^{2}10^{-1}$
$N=10$ , $k=1$
$1^{-1}2^{2}5^{-1}10^{2}$	$1^{-1}2^{2}5^{0}10^{1}$	$1^{-1}2^{2}5^{1}10^{0}$	$1^{-1}2^{2}5^{2}10^{-1}$	$1^{0}2^{0}5^{0}10^{2}$	$1^{0}2^{0}5^{1}10^{1}$
$1^{0}2^{1}5^{-1}10^{2}$	$1^{0}2^{1}5^{0}10^{1}$	$1^{0}2^{1}5^{1}10^{0}$	$1^{0}2^{1}5^{2}10^{-1}$	$1^{1}2^{0}5^{-1}10^{2}$	$1^{1}2^{0}5^{0}10^{1}$
$1^{1}2^{0}5^{2}10^{-1}$	$1^{2}2^{-1}5^{-1}10^{2}$	$1^{2}2^{-1}5^{0}10^{1}$	$1^{2}2^{-1}5^{1}10^{0}$	$1^{2}2^{-1}5^{2}10^{-1}$
$N=10$ , $k=3/2$
$1^{-1}2^{2}5^{0}10^{2}$	$1^{-1}2^{2}5^{1}10^{1}$	$1^{-1}2^{2}5^{2}10^{0}$	$1^{0}2^{1}5^{0}10^{2}$	$1^{0}2^{1}5^{1}10^{1}$	$1^{0}2^{1}5^{2}10^{0}$
$1^{0}2^{2}5^{-1}10^{2}$	$1^{0}2^{2}5^{0}10^{1}$	$1^{0}2^{2}5^{1}10^{0}$	$1^{0}2^{2}5^{2}10^{-1}$	$1^{1}2^{0}5^{0}10^{2}$	$1^{1}2^{0}5^{1}10^{1}$
$1^{1}2^{1}5^{-1}10^{2}$	$1^{1}2^{1}5^{0}10^{1}$	$1^{1}2^{1}5^{1}10^{0}$	$1^{1}2^{1}5^{2}10^{-1}$	$1^{2}2^{-1}5^{0}10^{2}$	$1^{2}2^{-1}5^{1}10^{1}$
$1^{2}2^{-1}5^{2}10^{0}$	$1^{2}2^{0}5^{-1}10^{2}$	$1^{2}2^{0}5^{0}10^{1}$	$1^{2}2^{0}5^{2}10^{-1}$
$N=10$ , $k=2$
$1^{0}2^{2}5^{1}10^{1}$	$1^{0}2^{2}5^{2}10^{0}$	$1^{1}2^{1}5^{0}10^{2}$	$1^{1}2^{1}5^{1}10^{1}$	$1^{1}2^{1}5^{2}10^{0}$	$1^{2}2^{0}5^{0}10^{2}$
$1^{2}2^{0}5^{1}10^{1}$
$N=11$ , $k=3/2$
$1^{0}11^{3}$
$N=11$ , $k=2$
$1^{0}11^{4}$	$1^{1}11^{3}$	$1^{3}11^{1}$
$N=11$ , $k=5/2$
$1^{1}11^{4}$	$1^{4}11^{1}$
$N=13$ , $k=1/2$
$1^{0}13^{1}$
$N=13$ , $k=1$
$1^{1}13^{1}$
$N=14$ , $k=1$
$1^{0}2^{0}7^{-2}14^{4}$	$1^{0}2^{0}7^{-1}14^{3}$	$1^{0}2^{0}7^{0}14^{2}$	$1^{0}2^{0}7^{3}14^{-1}$	$1^{0}2^{0}7^{4}14^{-2}$
$N=14$ , $k=3/2$
$1^{-2}2^{4}7^{0}14^{1}$	$1^{-2}2^{4}7^{1}14^{0}$	$1^{-2}2^{4}7^{2}14^{-1}$	$1^{-1}2^{2}7^{2}14^{0}$	$1^{-1}2^{2}7^{3}14^{-1}$	$1^{-1}2^{2}7^{4}14^{-2}$
$1^{-1}2^{3}7^{0}14^{1}$	$1^{-1}2^{3}7^{1}14^{0}$	$1^{-1}2^{3}7^{2}14^{-1}$	$1^{0}2^{0}7^{0}14^{3}$	$1^{0}2^{1}7^{-2}14^{4}$	$1^{0}2^{1}7^{-1}14^{3}$
$1^{0}2^{1}7^{0}14^{2}$	$1^{0}2^{1}7^{2}14^{0}$	$1^{0}2^{1}7^{3}14^{-1}$	$1^{0}2^{1}7^{4}14^{-2}$	$1^{0}2^{2}7^{0}14^{1}$	$1^{0}2^{2}7^{1}14^{0}$
$1^{0}2^{2}7^{2}14^{-1}$	$1^{1}2^{0}7^{-2}14^{4}$	$1^{1}2^{0}7^{-1}14^{3}$	$1^{1}2^{0}7^{0}14^{2}$	$1^{1}2^{0}7^{3}14^{-1}$	$1^{1}2^{0}7^{4}14^{-2}$
$1^{2}2^{-1}7^{-2}14^{4}$	$1^{2}2^{-1}7^{-1}14^{3}$	$1^{2}2^{-1}7^{0}14^{2}$	$1^{2}2^{0}7^{-1}14^{2}$	$1^{2}2^{0}7^{0}14^{1}$	$1^{3}2^{-1}7^{-1}14^{2}$
$1^{3}2^{-1}7^{0}14^{1}$	$1^{3}2^{-1}7^{1}14^{0}$	$1^{4}2^{-2}7^{-1}14^{2}$	$1^{4}2^{-2}7^{0}14^{1}$	$1^{4}2^{-2}7^{1}14^{0}$
$N=14$ , $k=2$
$1^{0}2^{1}7^{0}14^{3}$	$1^{0}2^{1}7^{3}14^{0}$	$1^{0}2^{3}7^{0}14^{1}$	$1^{0}2^{3}7^{1}14^{0}$	$1^{1}2^{0}7^{0}14^{3}$	$1^{3}2^{0}7^{0}14^{1}$
$N=15$ , $k=1$
$1^{0}3^{0}5^{-1}15^{3}$	$1^{0}3^{0}5^{0}15^{2}$	$1^{0}3^{0}5^{3}15^{-1}$
$N=15$ , $k=3/2$
$1^{-1}3^{3}5^{0}15^{1}$	$1^{-1}3^{3}5^{1}15^{0}$	$1^{0}3^{0}5^{0}15^{3}$	$1^{0}3^{1}5^{-1}15^{3}$	$1^{0}3^{1}5^{0}15^{2}$	$1^{0}3^{1}5^{2}15^{0}$
$1^{0}3^{1}5^{3}15^{-1}$	$1^{0}3^{2}5^{0}15^{1}$	$1^{0}3^{2}5^{1}15^{0}$	$1^{1}3^{0}5^{-1}15^{3}$	$1^{1}3^{0}5^{0}15^{2}$	$1^{1}3^{0}5^{3}15^{-1}$
$1^{2}3^{0}5^{0}15^{1}$	$1^{3}3^{-1}5^{0}15^{1}$	$1^{3}3^{-1}5^{1}15^{0}$
$N=15$ , $k=2$
$1^{0}3^{1}5^{3}15^{0}$	$1^{0}3^{3}5^{1}15^{0}$	$1^{1}3^{0}5^{0}15^{3}$	$1^{3}3^{0}5^{0}15^{1}$
$N=17$ , $k=1$
$1^{0}17^{2}$
$N=17$ , $k=3/2$
$1^{1}17^{2}$	$1^{2}17^{1}$
$N=19$ , $k=1$
$1^{0}19^{2}$
$N=19$ , $k=3/2$
$1^{1}19^{2}$	$1^{2}19^{1}$
$N=21$ , $k=1$
$1^{0}3^{0}7^{0}21^{2}$
$N=21$ , $k=3/2$
$1^{0}3^{1}7^{2}21^{0}$	$1^{0}3^{2}7^{1}21^{0}$	$1^{1}3^{0}7^{0}21^{2}$	$1^{2}3^{0}7^{0}21^{1}$
$N=27$ , $k=1/2$
$1^{0}3^{0}9^{0}27^{1}$
$N=27$ , $k=1$
$1^{0}3^{0}9^{1}27^{1}$	$1^{0}3^{1}9^{0}27^{1}$	$1^{0}3^{2}9^{-1}27^{1}$
$N=27$ , $k=3/2$
$1^{0}3^{0}9^{2}27^{1}$	$1^{0}3^{1}9^{1}27^{1}$	$1^{0}3^{2}9^{0}27^{1}$
$N=27$ , $k=2$
$1^{0}3^{2}9^{1}27^{1}$

Remark 9.5.

Each entry in Table LABEL:table:admissibleTypeI is different from others. For instance, although the eta-quotient $\eta(\tau)^{3}$ , when considered as $\eta(\tau)^{3}\eta(2\tau)^{0}$ , generates the one-dimensional space $M_{3/2}(\Gamma_{0}(2),\chi)$ where $\chi\colon\widetilde{\Gamma_{0}(3)}\rightarrow\mathbb{C}^{\times}$ is the character of $\eta(\tau)^{3}\eta(2\tau)^{0}$ , we list $\eta(\tau)^{3}$ only in the group $N=1$ , not $N=2,k=3/2$ to avoid repetition. As a comparison, Tables 2 and 3 of [20], whose entries constitute a subset of Table LABEL:table:admissibleTypeI above, contain repeated eta-quotients. For example, $\eta(\tau)^{-1}\eta(2\tau)^{2}=\eta(\tau)^{-1}\eta(2\tau)^{2}\eta(4\tau)^{0}$ is listed both in the groups corresponding to $N=2,k=1/2$ and $N=4,k=1/2$ there.

Remark 9.6.

One should notice, e.g., that the formal product $1^{0}3^{0}5^{0}15^{2}$ which represents $\eta(15\tau)^{2}$ is in Table LABEL:table:admissibleTypeI but $1^{0}3^{0}5^{0}15^{1}$ is not. This is because $\dim_{\mathbb{C}}M_{1/2}(\Gamma_{0}(15),\chi)$ can not be computed using Theorem 4.2 (with $t=0$ ) for any $\chi$ but $\dim_{\mathbb{C}}M_{1}(\Gamma_{0}(15),\chi)$ can with $\chi$ being the character of $\eta(15\tau)^{2}$ as one may directly see by comparing both sides of the first inequality of (21). See also Table 1 and Table LABEL:table:wt1. Nevertheless, we can prove that $\eta(15\tau)$ is weakly admissible, as the following proposition shows.

Proposition 9.7.

Let $f$ be an eta-quotient (of integral exponents). If there is a positive integer $n$ such that $f^{n}$ is weakly admissible, then $f$ is weakly admissible. As a consequence, The $m$ -th root, where $m\in\mathbb{Z}_{\geq 1}$ , of an eta-quotient $\prod\eta(n\tau)^{r_{n}}$ in Table LABEL:table:admissibleTypeI is weakly admissible provided that $m\mid r_{n}$ for all $n$ .

Proof.

Let $N$ , $k$ be the level and weight of $f$ respectively. Let $\chi\colon\widetilde{\Gamma_{0}(N)}\rightarrow\mathbb{C}^{\times}$ be the character of $f$ . Then $\chi^{n}$ is the character¹³¹³13Here $\chi^{n}$ means the character on $\widetilde{\Gamma_{0}(N)}$ that sends $\left(\gamma,\varepsilon\right)$ to $\chi\left(\gamma,\varepsilon\right)^{n}$ where $\gamma\in\Gamma_{0}(N)$ and $\varepsilon\in\{\pm 1\}$ . of $f^{n}$ . First suppose $f^{n}$ is weakly admissible of type I, which means $f^{n}$ is holomorphic at all cusps and $\dim_{\mathbb{C}}M_{nk}(\Gamma_{0}(N),\chi^{n})=1$ . Then obviously $f$ is holomorphic at all cusps so $f\in M_{k}(\Gamma_{0}(N),\chi)$ . To prove $\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(N),\chi)=1$ , let $0\neq g\in M_{k}(\Gamma_{0}(N),\chi)$ be arbitrary. Then $g^{n}\in M_{nk}(\Gamma_{0}(N),\chi^{n})$ and hence $g^{n}=c\cdot f^{n}$ for some $c\in\mathbb{C}^{\times}$ . It follows that $g=c_{1}\cdot f$ where $c_{1}$ is an $n$ -th root of $c$ since $\mathfrak{H}$ is connected and $f$ is holomorphic on $\mathfrak{H}$ . This proves that $f$ is weakly admissible of type I. The case of type II is proved in a similar manner with spaces such as $M_{k}(\Gamma_{0}(N),\chi)$ replaced by $S_{k}(\Gamma_{0}(N),\chi)$ . ∎

Remark 9.8.

It should be noted that if $f^{n}$ is weakly admissible of type II, then we can only assert that $f$ is weakly admissible; we can say nothing about whether $f$ is of type I or II. For instance, consider $f(\tau)=\eta(\tau)$ , $n=24$ .

9.3. Generalized double coset operators

We recall the theory of generalized double coset operators developed in [20, Section 3]. This is a generalization of Wohlfahrt’s extension of Hecke operators; c.f. [19]. Let $l$ , $N$ be positive integers. Let $\chi$ be the character (6) where $D=2$ and $r_{n}$ are integers. Let $(r_{n}^{\prime})_{n\mid N}$ be another sequence of integers such that $\sum_{n}r_{n}=\sum_{n}r_{n}^{\prime}$ and let $\chi^{\prime}$ be the character (6) with $r_{n}$ replaced by $r_{n}^{\prime}$ (again $D=2$ ). We define a formal expression (c.f. [20, eq. (28) and (30)])

(52)

T_{l}f(\tau)=l^{-\frac{k}{2}}\cdot\sum_{\begin{subarray}{c}{a\mid l}\\ {(N,a)=1}\end{subarray}}a^{k}\sum_{\begin{subarray}{c}{0\leq b<d}\\ {(a,b,d)=1}\end{subarray}}\chi\widetilde{\left(\begin{smallmatrix}{-Nb+ax}&{z}\\ {-Nd}&{y}\end{smallmatrix}\right)}^{-1}\chi^{\prime}\widetilde{\left(\begin{smallmatrix}{ay}&{by-dz}\\ {N}&{x}\end{smallmatrix}\right)}^{-1}f\left(\frac{a\tau+b}{d}\right)

where $k\in\frac{1}{2}\mathbb{Z}$ , $d=\frac{l}{a}$ and $x,y,z$ are any integers (depending on $a$ , $b$ and $d$ ) such that $(Nd,-Nb+ax)=1$ and $(-Nb+ax)y+Ndz=1$ . If the dependence on $k$ , $N$ , $\chi$ and $\chi^{\prime}$ is important, then we also denote $T_{l}$ by $T_{l;k,N,\chi,\chi^{\prime}}$ or $T_{l;\chi,\chi^{\prime}}$ .

Theorem 9.9.

For $f$ being a meromorphic modular form of weight $k\in\frac{1}{2}\mathbb{Z}$ for the group $\Gamma_{0}(N)$ with character $\chi$ , $T_{l}f$ is well-defined (i.e., independent of the choices of $x,y,z$ ) and is a meromorphic modular form of the same weight for the same group with character $\chi^{\prime}$ provided that the following three conditions hold:

(53)		$\displaystyle l\sum_{n\mid N}\frac{N}{n}r_{n}\equiv\sum_{n\mid N}\frac{N}{n}r_{n}^{\prime}\pmod{24},$
(54)		$\displaystyle\sum_{n\mid N}nr_{n}\equiv l\sum_{n\mid N}nr_{n}^{\prime}\pmod{24},$
(55)		$\displaystyle l^{2\lvert k^{\prime}\rvert}\cdot\prod_{2\nmid r_{n}-r_{n}^{\prime}}n\text{ is a perfect square,}$

where $k^{\prime}=\frac{1}{2}\sum_{n}r_{n}$ . Moreover, if the above conditions hold, then $T_{l}$ maps $M_{k}(\Gamma_{0}(N),\chi)$ into $M_{k}(\Gamma_{0}(N),\chi^{\prime})$ and $S_{k}(\Gamma_{0}(N),\chi)$ into $S_{k}(\Gamma_{0}(N),\chi^{\prime})$ respectively.

Proof.

This is a combination of the last assertion of [20, Theorem 3.9], [20, Proposition 3.2(4),(5)] and [20, eq. (28), (30)]. ∎

Theorem 9.10.

Let $f_{1}=\prod_{n}\eta(n\tau)^{r_{n}}$ and $f_{2}=\prod_{n}\eta(n\tau)^{r^{\prime}_{n}}$ be weakly admissible eta-quotients of the same type (e.g., functions listed in Table LABEL:table:admissibleTypeI) of the same weight $k$ and the same level $N$ . Let $\chi$ and $\chi^{\prime}$ be the characters of $f_{1}$ and $f_{2}$ on $\widetilde{\Gamma_{0}(N)}$ respectively. Then for any $l\in\mathbb{Z}_{\geq 1}$ satisfying (53), (54) and (55) (with $k^{\prime}=k$ ), there exists a $c_{l}\in\mathbb{C}$ such that

T_{l;\chi,\chi^{\prime}}f_{1}=c_{l}\cdot f_{2}.

Proof.

An immediate consequence of Theorem 9.9. ∎

Remark 9.11.

This theorem is still valid in the following two cases:

•

$f_{1}$ is weakly admissible of type II and $f_{2}$ is weakly admissible of type I,
•

$f_{1}$ is a cusp form that is weakly admissible of type I and $f_{2}$ is weakly admissible of type II.

Remark 9.12.

In the case $N=1$ (hence $r_{1}=r^{\prime}_{1}$ , $k=\frac{1}{2}r_{1}$ and $\chi=\chi^{\prime}$ ) the identity reads $T_{l}\eta^{r_{1}}=c_{l}\cdot\eta^{r_{1}}$ where $r_{1}=0,1,2,\dots 24$ . This has been investigated in detail in [20, Section 5] where $c_{l}$ and (52) are given in explicit forms.

We are mainly interested in the case $f_{1}=f_{2}$ in Theorem 9.10.

Corollary 9.13.

Let $f(\tau)=\prod_{n}\eta(n\tau)^{r_{n}}$ be a weakly admissible eta-quotient (e.g., a function listed in Table LABEL:table:admissibleTypeI) of weight $k=\frac{1}{2}\sum_{n}r_{n}$ and level $N$ . We define a submonoid of the multiplicative monoid of positive integers as follows:

L_{f}=\begin{dcases}\left\{l\in\mathbb{Z}_{\geq 1}\colon l\equiv 1\bmod{m_{f}}\right\}&\text{ if }k\in\mathbb{Z},\\ \left\{l\in\mathbb{Z}_{\geq 1}\colon l\equiv 1\bmod{m_{f}}\text{ and }l\text{ is a perfect square}\right\}&\text{ if }k\in\frac{1}{2}+\mathbb{Z},\end{dcases}

where

m_{f}=\frac{24}{(24,\sum_{n}(N/n)r_{n},\sum_{n}nr_{n})}.

Then for any $l\in L_{f}$ we have $T_{l}f=c_{l}\cdot f$ for some $c_{l}\in\mathbb{C}$ .

Proof.

Elementary number theory shows that $l\in L_{f}$ if and only if (53), (54) and (55) with $r^{\prime}_{n}=r_{n}$ and $k^{\prime}=k$ hold. Therefore the assertion follows from Theorem 9.10 with $f_{1}=f_{2}=f$ . ∎

Remark 9.14.

The monoid $L_{f}$ and the integer $m_{f}$ actually depend only on $\mathbf{r}=(r_{n})_{n\mid N}$ , so we sometimes write $L_{\mathbf{r}}$ and $m_{\mathbf{r}}$ instead.

In another words, any weakly admissible eta-quotient $f$ is an eigenform for the operators $T_{l}$ indexed by the infinite monoid $L_{f}$ . When $k\in\mathbb{Z}$ and the character of $f$ is induced by some Dirichlet character, then by a result of Gordon, Hughes and Newman (cf. [39, Theorem 1.64]) we have $m_{f}=1$ and hence $L_{f}=\mathbb{Z}_{\geq 1}$ . Moreover, in this case, $T_{l}$ is proportional to the usual Hecke operator of index $l$ . Consequently, the functions in Table LABEL:table:admissibleTypeI are all Hecke eigenforms for appropriate infinite operator monoids in a generalized sense.

The identities $T_{l}f=c_{l}\cdot f$ become interesting only when we can express (52) in a more explicit form, at least one where $x,y,z$ do not appear. The following theorem gives such a formula which is key to the current topic. The reader may compare this with a previous result of Wohlfahrt [19, eq. (7 $|$ 3) and (10 $|$ 3)].

Theorem 9.15.

Let $N\in\mathbb{Z}_{\geq 1}$ . For each $n\mid N$ , let $r_{n}$ be an integer and set

(56)

\mathbf{r}=(r_{n})_{n\mid N},\quad k^{\prime}=\frac{1}{2}\sum_{n\mid N}r_{n},\quad x_{N}=\sum_{n\mid N}nr_{n},\quad\varPi=\prod_{n\mid N}(Nn^{-1})^{\lvert r_{n}\rvert}.

Let $\chi_{\mathbf{r}}\colon\widetilde{\Gamma_{0}(N)}\rightarrow\mathbb{C}^{\times}$ be the character of $\prod_{n\mid N}\eta(n\tau)^{r_{n}}$ , i.e. (6). Let $f$ be a meromorphic modular form on $\Gamma_{0}(N)$ of weight $k\in k^{\prime}+2\mathbb{Z}$ with character $\chi_{\mathbf{r}}$ and let $l\in L_{\mathbf{r}}$ . Then

(a)

we have the expansions (on $\Im\tau>Y_{0}$ )

f(\tau)=\sum_{n\in\frac{x_{N}}{24}+\mathbb{Z}}c_{f}(n)q^{n},\quad T_{l}f(\tau)=\sum_{n\in\frac{x_{N}}{24}+\mathbb{Z}}c_{T_{l}f}(n)q^{n},

where $Y_{0}$ is some nonnegative number, $c_{f}(n),\,c_{T_{l}f}(n)\in\mathbb{C}$ are uniquely determined and $T_{l}$ is the operator defined in (52) with $\chi=\chi^{\prime}=\chi_{\mathbf{r}}$ ;

(b)

for $n\in\frac{x_{N}}{24}+\mathbb{Z}$ we have

(57)

c_{T_{l}f}(n)=l^{-\frac{k}{2}}\sum_{\begin{subarray}{c}{a\mid l,\,d=l/a}\\ {(a,N)=1}\end{subarray}}\genfrac{(}{)}{}{}{a}{\varPi}a^{k}\cdot c_{f}\left(\frac{ln}{a^{2}}\right)\sum_{\begin{subarray}{c}{0\leq b<d}\\ {(a,b,d)=1}\end{subarray}}\genfrac{(}{)}{}{}{-Nb}{(a,d)}^{2k}\mathfrak{e}\left(bd\left(\frac{n}{l}-\frac{x_{N}}{24}\right)\right)\psi_{l,\mathbf{r}}(a,b)

where

\psi_{l,\mathbf{r}}(a,b)=\begin{dcases}\mathfrak{e}\left(-\frac{k+\delta}{4}(d-1)+\frac{(k+\delta)(l-1)(N-1)}{4}\right)&\text{ if }2\nmid l,\\ \mathfrak{e}\left(-\frac{k+\delta}{4}(a-1)-\frac{N(k+\delta)(1+\delta_{1})}{4}b\right)&\text{ if }2\mid l,\,2\mid N,\\ 1&\text{ if }2\mid l,\,2\nmid N,\end{dcases}

with $\delta=0$ ( $\delta=1$ respectively) if $\varPi$ takes the form $2^{\alpha}\cdot(4m+1)$ ( $2^{\alpha}\cdot(4m+3)$ respectively), $\alpha,\,m\in\mathbb{Z}_{\geq 0}$ , $\delta_{1}=1$ if $2\mid l$ , $4\mid N$ , $k\in\frac{1}{2}+\mathbb{Z}$ , $v_{2}(\varPi)\equiv 1\bmod{2}$ and $\delta_{1}=0$ otherwise.

A few words of caution are in order. Note that $\genfrac{(}{)}{}{}{a}{\varPi}$ and $\genfrac{(}{)}{}{}{-Nb}{(a,d)}$ refer to Kronecker-Jacobi symbols; they are not fractions. Also note that in (57) we have set $c_{f}\left(\frac{ln}{a^{2}}\right)=0$ if $\frac{ln}{a^{2}}\not\in\frac{x_{N}}{24}+\mathbb{Z}$ . In addition, $k$ can be replaced by $k^{\prime}$ in the definition of $\psi_{l,\mathbf{r}}(a,b)$ and if $f$ is the eta-quotient $\prod_{n\mid N}\eta(n\tau)^{r_{n}}$ itself, which is the case we will use below, then $k=k^{\prime}$ . Finally, $Y_{0}=0$ if $f$ is holomorphic on $\mathfrak{H}$ .

Remark 9.16.

The formula (57) is equivalent to

(58)

T_{l}f(\tau)=l^{-\frac{k}{2}}\sum_{\begin{subarray}{c}{a\mid l,\,d=l/a}\\ {(a,N)=1}\end{subarray}}\genfrac{(}{)}{}{}{a}{\varPi}a^{k}\sum_{\begin{subarray}{c}{0\leq b<d}\\ {(a,b,d)=1}\end{subarray}}\genfrac{(}{)}{}{}{-Nb}{(a,d)}^{2k}\mathfrak{e}\left(-\frac{x_{N}}{24}bd\right)\psi_{l,\mathbf{r}}(a,b)\cdot f\left(\frac{a\tau+b}{d}\right)

which can be considered as an explicit form of (52) with $\chi=\chi^{\prime}$ .

Proof of Theorem 9.15 and Remark 9.16.

We begin with the proof of (58). Set

R:=\chi\widetilde{\left(\begin{smallmatrix}{-Nb+ax}&{z}\\ {-Nd}&{y}\end{smallmatrix}\right)}\chi^{\prime}\widetilde{\left(\begin{smallmatrix}{ay}&{by-dz}\\ {N}&{x}\end{smallmatrix}\right)}=\chi_{\mathbf{r}}\widetilde{\left(\begin{smallmatrix}{(-Nb+ax)ay+Nz}&{(-Nb+ax)(by-dz)+xz}\\ {-Nady+Ny}&{-Nd(by-dz)+xy}\end{smallmatrix}\right)}

in (52). (Theorem 9.9 and the assumption $l\in L_{\mathbf{r}}$ ensure that $T_{l}$ is well-defined.) Since $ad=l$ and

(59)

(-Nb+ax)y+Ndz=1

we have

R=\chi_{\mathbf{r}}\widetilde{\left(\begin{smallmatrix}{a-N(l-1)z}&{b-(l-1)xz}\\ {-N(l-1)y}&{d-(l-1)xy}\end{smallmatrix}\right)}.

To derive (58) from (52), we need only to prove

(60)

R^{-1}=\genfrac{(}{)}{}{}{a}{\varPi}\genfrac{(}{)}{}{}{-Nb}{(a,d)}^{2k}\mathfrak{e}\left(-\frac{x_{N}}{24}bd\right)\psi_{l,\mathbf{r}}(a,b)

for which we need the Petersson’s formula of $\chi_{\eta}$ :

(61)

\chi_{\eta}\left(\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right),\varepsilon\right)=\begin{cases}\varepsilon\cdot\genfrac{(}{)}{}{}{d}{\lvert c\rvert}\mathfrak{e}\left(\frac{1}{24}\left((a-2d)c-bd(c^{2}-1)+(3d-3)c\right)\right)&\text{if }2\nmid c,\\ \varepsilon\cdot\genfrac{(}{)}{}{}{c}{d}\mathfrak{e}\left(\frac{1}{24}\left((a-2d)c-bd(c^{2}-1)+3d-3\right)\right)&\text{if }2\mid c.\end{cases}

For a proof of (61) see [42]. Since $r_{n}\in\mathbb{Z}$ we have

(62)

R=\prod_{n\mid N}\chi_{\eta}^{r_{n}}\widetilde{\left(\begin{smallmatrix}{a-N(l-1)z}&{n(b-(l-1)xz)}\\ {-\frac{N}{n}(l-1)y}&{d-(l-1)xy}\end{smallmatrix}\right)}.

The strategy¹⁴¹⁴14The $N=1$ case has been successfully worked out in the proof of [20, Lemma 5.2]. The strategy here is essentially the one there. is to insert (61) into (62) and then eliminate $x,\,y,\,z$ using properties of Kronecker-Jacobi symbols. Inserting (61) into (62) gives

(63)

R=\prod_{\begin{subarray}{c}{n\mid N}\\ {2\nmid\frac{N}{n}(l-1)y}\end{subarray}}\genfrac{(}{)}{}{}{d-(l-1)xy}{Nn^{-1}(l-1)\lvert y\rvert}^{r_{n}}\cdot\prod_{\begin{subarray}{c}{n\mid N}\\ {2\mid\frac{N}{n}(l-1)y}\end{subarray}}\genfrac{(}{)}{}{}{-Nn^{-1}(l-1)y}{d-(l-1)xy}^{r_{n}}\cdot\mathfrak{e}\left(\frac{x_{N}}{24}bd\right)\\ \cdot\prod_{\begin{subarray}{c}{n\mid N}\\ {2\nmid\frac{N}{n}(l-1)y}\end{subarray}}\mathfrak{e}\left(\frac{r_{n}}{8}(d-1-(l-1)xy)(-Nn^{-1})(l-1)y\right)\cdot\prod_{\begin{subarray}{c}{n\mid N}\\ {2\mid\frac{N}{n}(l-1)y}\end{subarray}}\mathfrak{e}\left(\frac{r_{n}}{8}(d-1-(l-1)xy)\right)

where we have used the fact

(64)

24\mid(l-1)\sum_{n\mid N}nr_{n},\quad 24\mid(l-1)\sum_{n\mid N}Nn^{-1}r_{n}

which are consequences of the assumption $l\in L_{\mathbf{r}}$ . Now the proof splits into three cases.

The case $2\nmid l$ . We must have $2\mid\frac{N}{n}(l-1)y$ for any $n\mid N$ . Therefore

(65)

R=\mathfrak{e}\left(\frac{x_{N}}{24}bd\right)\prod_{n\mid N}\genfrac{(}{)}{}{}{-Nn^{-1}(l-1)y}{d-(l-1)xy}^{r_{n}}\prod_{n\mid N}\mathfrak{e}\left(\frac{r_{n}}{8}(d-1-(l-1)xy)\right).

Note that if $k^{\prime}\in\frac{1}{2}+\mathbb{Z}$ then $l$ is an odd square and hence

\genfrac{(}{)}{}{}{-Nn^{-1}(l-1)y}{d-(l-1)xy}=\genfrac{(}{)}{}{}{Nn^{-1}}{d-(l-1)xy}\cdot\genfrac{(}{)}{}{}{-(l-1)y}{d-(l-1)xy}=\genfrac{(}{)}{}{}{Nn^{-1}}{d-(l-1)xy}\cdot\genfrac{(}{)}{}{}{y}{d}.

Inserting this into (65) we find that

(66)

R=\mathfrak{e}\left(\frac{x_{N}}{24}bd\right)\cdot\genfrac{(}{)}{}{}{\varPi}{d-(l-1)xy}\genfrac{(}{)}{}{}{y}{d}^{2k^{\prime}}\mathfrak{e}\left(\frac{k^{\prime}}{4}(d-1-(l-1)xy)\right).

If $k^{\prime}\in\mathbb{Z}$ , the above expression is still valid. By (59) if $k^{\prime}\in\frac{1}{2}+\mathbb{Z}$ we have

(67)

\genfrac{(}{)}{}{}{y}{d}=\genfrac{(}{)}{}{}{-Nb+ax}{d}=\genfrac{(}{)}{}{}{-Nb}{(a,d)}.

For the last equality, see [20, p. 567, line -3]. If $k^{\prime}\in\mathbb{Z}$ , then trivially $\genfrac{(}{)}{}{}{y}{d}^{2k^{\prime}}=\genfrac{(}{)}{}{}{-Nb}{(a,d)}^{2k^{\prime}}$ . For the calculation of $\genfrac{(}{)}{}{}{\varPi}{d-(l-1)xy}$ , we set

(68)

\varPi^{\prime}=\prod_{p\mid\varPi,\,2\nmid v_{p}(\varPi)}p.

There are four cases to consider. For the first case, $\varPi^{\prime}\equiv 1\bmod{4}$ , we have

(69)

\genfrac{(}{)}{}{}{\varPi}{d-(l-1)xy}=\genfrac{(}{)}{}{}{\varPi^{\prime}}{d-(l-1)xy}=\genfrac{(}{)}{}{}{d-(l-1)xy}{\varPi^{\prime}}=\genfrac{(}{)}{}{}{a}{\varPi^{\prime}}=\genfrac{(}{)}{}{}{a}{\varPi}.

The third equality above follows from

\genfrac{(}{)}{}{}{d-(l-1)xy}{p}=\genfrac{(}{)}{}{}{d-lxy+xy}{p}=\genfrac{(}{)}{}{}{xy}{p}=\genfrac{(}{)}{}{}{a}{p},\quad p\mid\varPi^{\prime}

since by (59) we have $axy\equiv 1\bmod{p}$ and $lxy\equiv d\bmod{p}$ . Moreover, we have

(70)

\mathfrak{e}\left(-\frac{k^{\prime}(l-1)xy}{4}\right)=\mathfrak{e}\left(-\frac{k^{\prime}(l-1)(N-1)}{4}\right).

(When $k^{\prime}\in\mathbb{Z}$ , $l\equiv 3\bmod{4}$ and $2\nmid N$ , (70) does not necessarily hold, but we can always choose $x,y,z$ such that $2\mid y$ for which (70) holds. We always make such a choice.) Inserting (67), (69) and (70) into (66) and replacing $k^{\prime}$ by $k$ we obtain (60) as required. For the other three cases $\varPi^{\prime}\equiv 3\bmod{4}$ , $\varPi^{\prime}\equiv 2\bmod{8}$ and $\varPi^{\prime}\equiv 6\bmod{8}$ , we proceed similarly and obtain (60) likewise. However, if $2\mid\varPi^{\prime}$ there is an extra difficulty: a factor $\genfrac{(}{)}{}{}{2}{ad-(l-1)axy}=\genfrac{(}{)}{}{}{2}{1+(l-1)N(dz-by)}$ appears and we should prove it equals $1$ . This is immediate if $l\equiv 1\bmod{4}$ since $2\mid\varPi^{\prime}$ implies $2\mid N$ . Now suppose $l\equiv 3\bmod{4}$ . Assume by contradiction that $4\nmid N$ ; we have $N\equiv 2\bmod{4}$ . By the first relation of (64) we have $4\mid\sum_{n}nr_{n}$ from which $2\mid\sum_{2\nmid n}r_{n}$ . Since $2\mid\varPi^{\prime}$ we have

v_{2}(\varPi)=\sum_{n\mid N}v_{2}(N/n)\cdot\lvert r_{n}\rvert=\sum_{2\nmid n}\lvert r_{n}\rvert\equiv 1\bmod{2}

which contradicts $2\mid\sum_{2\nmid n}r_{n}$ . Therefore, $4\mid N$ and hence $\genfrac{(}{)}{}{}{2}{1+(l-1)N(dz-by)}=1$ .

The case $2\mid l$ , $2\nmid N$ . By (59) if $2\mid d$ then $2\nmid y$ and if $2\nmid d$ then we can choose $x,y,z$ such that $2\nmid y$ . (If $(x,y,z)$ is a solution to (59) with $2\mid y$ , then $(x,y+Nd,z-(-Nb+ax))$ is another solution with $2\nmid y+Nd$ .) We always make such a choice. Thus (63) becomes

R=\mathfrak{e}\left(\frac{x_{N}}{24}bd\right)\prod_{n\mid N}\genfrac{(}{)}{}{}{d-(l-1)xy}{Nn^{-1}(l-1)\lvert y\rvert}^{r_{n}}\prod_{n\mid N}\mathfrak{e}\left(\frac{r_{n}}{8}(d-1-(l-1)xy)(-Nn^{-1})(l-1)y\right).

Since $l\equiv 1\bmod{m_{\mathbf{r}}}$ we have $2\nmid m_{\mathbf{r}}$ . By the definition of $m_{\mathbf{r}}$ (c.f. Remark 9.14) we have

(71)

8\mid\sum_{n}nr_{n},\quad 8\mid\sum_{n}Nn^{-1}r_{n}.

It follows that $k^{\prime}=\frac{1}{2}\sum_{n}r_{n}\in\mathbb{Z}$ and

(72)

R=\mathfrak{e}\left(\frac{x_{N}}{24}bd\right)\prod_{n\mid N}\genfrac{(}{)}{}{}{d-(l-1)xy}{Nn^{-1}(l-1)\lvert y\rvert}^{r_{n}}.

Note that

\genfrac{(}{)}{}{}{d-(l-1)xy}{Nn^{-1}(l-1)\lvert y\rvert}=\genfrac{(}{)}{}{}{d-(l-1)xy}{Nn^{-1}}\cdot\genfrac{(}{)}{}{}{d-(l-1)xy}{(l-1)\lvert y\rvert}=\genfrac{(}{)}{}{}{d-(l-1)xy}{Nn^{-1}}\cdot\genfrac{(}{)}{}{}{d}{(l-1)\lvert y\rvert}

where the last equality follows from the fact $(l-1)\lvert y\rvert$ is odd and positive. Inserting this and $2k^{\prime}\equiv 0\bmod{2}$ into (72) we obtain

R=\mathfrak{e}\left(\frac{x_{N}}{24}bd\right)\cdot\genfrac{(}{)}{}{}{d-(l-1)xy}{\varPi}=\mathfrak{e}\left(\frac{x_{N}}{24}bd\right)\cdot\genfrac{(}{)}{}{}{a}{\varPi}

where the last equality follows just as in the deduction of (69). We thus have arrived at (60).

The case $2\mid l$ , $2\mid N$ . As in the last case, we have $8\mid\sum_{n}nr_{n}$ and $8\mid\sum_{n}Nn^{-1}r_{n}$ . Moreover, $y$ must be odd by (59). For any $n\mid N$ with $2\nmid\frac{N}{n}(l-1)y$ , that is, with $2\nmid\frac{N}{n}$ , we have

\genfrac{(}{)}{}{}{d-(l-1)xy}{Nn^{-1}(l-1)\lvert y\rvert}=\genfrac{(}{)}{}{}{Nn^{-1}(l-1)\lvert y\rvert}{d-(l-1)xy}\cdot\mathfrak{e}\left(-\frac{1}{8}(d-1-(l-1)xy)\cdot(Nn^{-1}(l-1)\lvert y\rvert-1)\right).

Choosing a solution $(x,y,z)$ to (59) with $y<0$ and inserting the above identity into (63), we find that

R=\mathfrak{e}\left(\frac{x_{N}}{24}bd\right)\cdot\prod_{n\mid N}\genfrac{(}{)}{}{}{-Nn^{-1}(l-1)y}{d-(l-1)xy}^{r_{n}}\cdot\prod_{\begin{subarray}{c}{n\mid N}\\ {2\nmid\frac{N}{n}}\end{subarray}}\mathfrak{e}\left(-\frac{r_{n}}{8}(d-1-(l-1)xy)(-Nn^{-1}(l-1)y-1)\right)\\ \cdot\prod_{\begin{subarray}{c}{n\mid N}\\ {2\nmid\frac{N}{n}}\end{subarray}}\mathfrak{e}\left(\frac{r_{n}}{8}(d-1-(l-1)xy)(-Nn^{-1})(l-1)y\right)\cdot\prod_{\begin{subarray}{c}{n\mid N}\\ {2\mid\frac{N}{n}}\end{subarray}}\mathfrak{e}\left(\frac{r_{n}}{8}(d-1-(l-1)xy)\right)

We deal with the factor $\prod_{n\mid N}\genfrac{(}{)}{}{}{-Nn^{-1}(l-1)y}{d-(l-1)xy}^{r_{n}}$ as in previous cases and thus simplify the expression for $R$ as

(73)

R=\mathfrak{e}\left(\frac{x_{N}}{24}bd\right)\cdot\genfrac{(}{)}{}{}{\varPi}{d-(l-1)xy}\cdot\genfrac{(}{)}{}{}{d}{-(l-1)y}^{2k^{\prime}}\cdot\mathfrak{e}\left(\frac{k^{\prime}}{4}(d-1-(l-1)xy)\cdot(-(l-1)y)\right).

There are six subcases: (c.f. (68) for $\varPi^{\prime}$ )

(a)

$2\mid l$ , $4\mid N$ , $k^{\prime}\in\mathbb{Z}$ and $2\nmid\varPi^{\prime}$ .
(b)

$2\mid l$ , $4\mid N$ , $k^{\prime}\in\mathbb{Z}$ and $2\mid\varPi^{\prime}$ .
(c)

$2\mid l$ , $4\mid N$ , $k^{\prime}\in\frac{1}{2}+\mathbb{Z}$ and $2\nmid\varPi^{\prime}$ .
(d)

$2\mid l$ , $4\mid N$ , $k^{\prime}\in\frac{1}{2}+\mathbb{Z}$ and $2\mid\varPi^{\prime}$ . (Only in this subcase we have $\delta_{1}=1$ .)
(e)

$2\mid l$ , $N\equiv 2\bmod{4}$ and $2\nmid\varPi^{\prime}$ . (We must have $k^{\prime}\in\mathbb{Z}$ in this subcase.)
(f)

$2\mid l$ , $N\equiv 2\bmod{4}$ and $2\mid\varPi^{\prime}$ . (We must have $k^{\prime}\in\mathbb{Z}$ in this subcase.)

We will only give the proof of the more difficult subcases (b) and (d) and omit the other four since the strategies are the same. For (b), we first prove $8\mid N$ . Assume by contradiction that $8\nmid N$ ; then $2^{2}\parallel N$ . It follows that

v_{2}(\varPi)\equiv\sum_{\begin{subarray}{c}{n\mid N}\\ {v_{2}(n)=1}\end{subarray}}v_{2}(N/n)\cdot\lvert r_{n}\rvert=\sum_{\begin{subarray}{c}{n\mid N}\\ {v_{2}(n)=1}\end{subarray}}\lvert r_{n}\rvert\equiv 1\bmod{2}.

On the other hand, (71) still holds in this case, and hence

\sum_{\begin{subarray}{c}{n\mid N}\\ {v_{2}(n)=0}\end{subarray}}r_{n}\equiv\sum_{\begin{subarray}{c}{n\mid N}\\ {v_{2}(n)=2}\end{subarray}}r_{n}\equiv 0\bmod{2}.

Therefore $\sum_{n\mid N}r_{n}\equiv 1\bmod{2}$ , that is, $k^{\prime}\in\frac{1}{2}+\mathbb{Z}$ , which contradicts the assumption in (b). Hence $8\mid N$ . Note that both $\varPi^{\prime}/2$ and $d-(l-1)xy$ are odd and positive, from which we deduce that

	$\displaystyle\genfrac{(}{)}{}{}{\varPi}{d-(l-1)xy}$	$\displaystyle=\genfrac{(}{)}{}{}{\varPi^{\prime}}{d-(l-1)xy}$
		$\displaystyle=\genfrac{(}{)}{}{}{2}{d-(l-1)xy}\genfrac{(}{)}{}{}{d-(l-1)xy}{\varPi^{\prime}/2}\mathfrak{e}\left(\frac{1}{4}(d-1-(l-1)xy)\cdot\frac{\varPi^{\prime}/2-1}{2}\right)$
		$\displaystyle=\genfrac{(}{)}{}{}{2}{ad-(l-1)axy}\genfrac{(}{)}{}{}{a}{\varPi^{\prime}}\mathfrak{e}\left(\frac{1}{4}(d-1-(l-1)xy)\cdot\frac{\varPi^{\prime}/2-1}{2}\right)$
		$\displaystyle=\genfrac{(}{)}{}{}{2}{1+(l-1)N(dz-by)}\genfrac{(}{)}{}{}{a}{\varPi}\mathfrak{e}\left(\frac{1}{4}(d-1-(l-1)xy)\cdot\frac{\varPi^{\prime}/2-1}{2}\right)$
(74)			$\displaystyle=\genfrac{(}{)}{}{}{a}{\varPi}\mathfrak{e}\left(\frac{1}{4}(a-1)\cdot\frac{\varPi^{\prime}/2-1}{2}\right).$

The last equality holds since by (59) and $8\mid N$ we have $axy\equiv 1\bmod{4}$ and hence

(a-1)-(d-1-(l-1)xy)\equiv a-d+(l-1)a=(a^{2}-1)d\equiv 0\bmod{4}.

An argument similar to the above deduction shows that

\mathfrak{e}\left(\frac{k^{\prime}}{4}(d-1-(l-1)xy)\cdot(-(l-1)y)\right)=\mathfrak{e}\left(\frac{k^{\prime}}{4}(a-1)(l-1)\right).

Inserting this and (74) into (73) and noting that $a$ is odd, we obtain (60) in the subcase (b). For the subcase (d), $l$ is a square. Since $(N,a)=1$ and $ad=l$ we have $v_{2}(d)=v_{2}(l)\equiv 0\bmod{2}$ and hence $4\mid d$ . Thus if we let $d^{\prime}=d/2^{v_{2}(d)}$ then

\genfrac{(}{)}{}{}{d}{-(l-1)y}=\genfrac{(}{)}{}{}{d}{y}=\genfrac{(}{)}{}{}{d^{\prime}}{y}=\mathfrak{e}\left(\frac{y-1}{4}\cdot\frac{d^{\prime}-1}{2}\right)\cdot\genfrac{(}{)}{}{}{y}{d}.

Combining this and

\genfrac{(}{)}{}{}{-Nb+ax}{d}\genfrac{(}{)}{}{}{y}{d}=\genfrac{(}{)}{}{}{(-Nb+ax)y}{d}=\genfrac{(}{)}{}{}{1-Ndz}{d}=1

we find that

(75)

\genfrac{(}{)}{}{}{d}{-(l-1)y}=\mathfrak{e}\left(\frac{y-1}{4}\cdot\frac{d^{\prime}-1}{2}\right)\cdot\genfrac{(}{)}{}{}{-Nb+ax}{d}=\mathfrak{e}\left(\frac{y-1}{4}\cdot\frac{d^{\prime}-1}{2}\right)\cdot\genfrac{(}{)}{}{}{-Nb}{(a,d)}.

The calculation of $\genfrac{(}{)}{}{}{\varPi}{d-(l-1)xy}$ is similar to that in above cases (e.g. (69) or (74)) and the result is

(76)

\genfrac{(}{)}{}{}{\varPi}{d-(l-1)xy}=\genfrac{(}{)}{}{}{2}{1+Nb}\genfrac{(}{)}{}{}{a}{\varPi}\mathfrak{e}\left(\frac{1}{4}(l-1)(a-1)\cdot\frac{\varPi^{\prime}/2-1}{2}\right).

For the factor $\mathfrak{e}\left(\frac{k^{\prime}}{4}(d-1-(l-1)xy)\cdot(-(l-1)y)\right)$ in (73), note that $l-d=(a-1)d\equiv 0\bmod{8}$ , $2\nmid y$ and hence

	$\displaystyle\mathfrak{e}\left(\frac{k^{\prime}}{4}(d-1-(l-1)xy)\cdot(-(l-1)y)\right)$	$\displaystyle=\mathfrak{e}\left(\frac{k^{\prime}}{4}(l-1)^{2}(1-xy)(-y)\right)$
		$\displaystyle=\mathfrak{e}\left(\frac{k^{\prime}}{4}(1-xy)(-y)\right)=\mathfrak{e}\left(\frac{k^{\prime}}{4}(x-y)\right).$

Inserting this equality, (75) and (76) into (73) we obtain

(77)

R=\genfrac{(}{)}{}{}{a}{\varPi}\genfrac{(}{)}{}{}{-Nb}{(a,d)}\mathfrak{e}\left(\frac{x_{N}}{24}bd\right)\genfrac{(}{)}{}{}{2}{1+Nb}\\ \cdot\mathfrak{e}\left(\frac{1}{4}(l-1)(a-1)\cdot\frac{\varPi^{\prime}/2-1}{2}\right)\mathfrak{e}\left(\frac{y-1}{4}\cdot\frac{d^{\prime}-1}{2}+\frac{k^{\prime}}{4}(x-y)\right).

Now we simplify the factor $\mathfrak{e}\left(\frac{y-1}{4}\cdot\frac{d^{\prime}-1}{2}+\frac{k^{\prime}}{4}(x-y)\right)$ in above. By (59) and the facts $2\mid d$ , $4\mid N$ we find that $(-Nb+ax)y\equiv 1\bmod{8}$ . This, together with $2\nmid y$ , implies $x\equiv ay+Nab\bmod{8}$ . It follows that

	$\displaystyle\mathfrak{e}\left(\frac{y-1}{4}\cdot\frac{d^{\prime}-1}{2}+\frac{k^{\prime}}{4}(x-y)\right)$	$\displaystyle=\mathfrak{e}\left(\frac{-2k^{\prime}}{4}(y-1)\cdot\frac{d^{\prime}-1}{2}+\frac{k^{\prime}}{4}(ay-y+Nab)\right)$
(78)			$\displaystyle=\mathfrak{e}\left(\frac{k^{\prime}}{4}Nab\right)\mathfrak{e}\left(\frac{k^{\prime}}{4}y(a-d^{\prime})\right)\mathfrak{e}\left(\frac{k^{\prime}}{4}(d^{\prime}-1)\right).$

Since $ad^{\prime}=l/2^{v_{2}(d)}=l/2^{v_{2}(l)}$ is a square, $2\nmid a$ and $2\nmid d^{\prime}$ we have

(79)

a-d^{\prime}\equiv 0\bmod{8}.

In addition, since $k^{\prime}\in\frac{1}{2}+\mathbb{Z}$ , $2\nmid a$ and $4\mid N$ ,

(80)

\mathfrak{e}\left(\frac{k^{\prime}}{4}Nab\right)=\mathfrak{e}\left(\frac{1}{8}Nab\right)=\mathfrak{e}\left(\frac{1}{8}Nb\right).

Inserting (79) and (80) into (78), then combining (77) and noting that $\mathfrak{e}\left(\frac{1}{8}Nb\right)\genfrac{(}{)}{}{}{2}{1+Nb}=1$ , we arrive at (60) as required. This concludes the proof of (58).

Finally, we begin to prove Theorem 9.15. For Theorem 9.15(a), elementary Fourier analysis shows that

f(\tau)=\sum_{n\in\frac{1}{24}\mathbb{Z}}c_{f}(n)q^{n},\quad T_{l}f(\tau)=\sum_{n\in\frac{1}{24}\mathbb{Z}}c_{T_{l}f}(n)q^{n},

which converge on $\Im\tau>Y_{0}$ for some $Y_{0}\geq 0$ since $f(\tau+24)=f(\tau)$ and $T_{l}f(\tau+24)=T_{l}f(\tau)$ . By modularity $f|_{k}\widetilde{\left(\begin{smallmatrix}{1}&{1}\\ {0}&{1}\end{smallmatrix}\right)}=\chi_{\mathbf{r}}\widetilde{\left(\begin{smallmatrix}{1}&{1}\\ {0}&{1}\end{smallmatrix}\right)}\cdot f=\mathfrak{e}\left(\frac{x_{N}}{24}\right)\cdot f$ and $T_{l}f|_{k}\widetilde{\left(\begin{smallmatrix}{1}&{1}\\ {0}&{1}\end{smallmatrix}\right)}=\chi_{\mathbf{r}}\widetilde{\left(\begin{smallmatrix}{1}&{1}\\ {0}&{1}\end{smallmatrix}\right)}\cdot T_{l}f=\mathfrak{e}\left(\frac{x_{N}}{24}\right)\cdot T_{l}f$ . In terms of Fourier coefficients,

c_{f}(n)\mathfrak{e}\left(n\right)=c_{f}(n)\mathfrak{e}\left(\frac{x_{N}}{24}\right),\quad c_{T_{l}f}(n)\mathfrak{e}\left(n\right)=c_{T_{l}f}(n)\mathfrak{e}\left(\frac{x_{N}}{24}\right).

It follows that $c_{f}(n)=0$ unless $n\in\frac{x_{N}}{24}+\mathbb{Z}$ and so is $c_{T_{l}f}(n)$ . Theorem 9.15(b), that is, (57), follows immediately from (58) whose details are omitted. ∎

In the important case $x_{N}\equiv 0\mod{24}$ we have $c_{f}(n)=0$ and $c_{T_{l}f}(n)=0$ unless $n\in\mathbb{Z}$ . Therefore, (57) becomes

(81)

c_{T_{l}f}(n)=l^{-\frac{k}{2}}\sum_{\begin{subarray}{c}{a\mid l,\,a^{2}\mid ln}\\ {(a,N)=1,\,d=l/a}\end{subarray}}\genfrac{(}{)}{}{}{a}{\varPi}a^{k}\cdot c_{f}\left(\frac{ln}{a^{2}}\right)\sum_{\begin{subarray}{c}{0\leq b<d}\\ {(a,b,d)=1}\end{subarray}}\genfrac{(}{)}{}{}{-Nb}{(a,d)}^{2k}\mathfrak{e}\left(\frac{bdn}{l}\right)\psi_{l,\mathbf{r}}(a,b)

for $n\in\mathbb{Z}$ . From this some elegant formulas for special cases follow:

Corollary 9.17.

We keep the notations and assumptions of Theorem 9.15 and assume that $x_{N}\equiv 0\mod{24}$ . Then

c_{T_{l}f}(0)=c_{f}(0)\cdot l^{-\frac{k}{2}}\sum_{\begin{subarray}{c}{a\mid l,\,d=l/a}\\ {(a,N)=1}\end{subarray}}\genfrac{(}{)}{}{}{a}{\varPi}a^{k}\sum_{\begin{subarray}{c}{0\leq b<d}\\ {(a,b,d)=1}\end{subarray}}\genfrac{(}{)}{}{}{-Nb}{(a,d)}^{2k}\psi_{l,\mathbf{r}}(a,b).

Moreover, if for any prime $p$ not dividing $N$ we have $p^{2}\nmid l$ (e.g. $l$ is square-free or $\mathop{\mathrm{rad}}(l)\mid\mathop{\mathrm{rad}}(N)$ ), then

(82)

c_{T_{l}f}(1)=c_{f}(l)\cdot l^{-\frac{k}{2}}\sum_{0\leq b<l}\psi_{l,\mathbf{r}}(1,b).

Finally, if $l=p^{\alpha}$ where $p$ is a prime not dividing $N$ and $\alpha\in\mathbb{Z}_{\geq 1}$ , then

c_{T_{p^{\alpha}}f}(1)=p^{-\frac{\alpha k}{2}}\sum_{0\leq\beta\leq[\alpha/2]}\genfrac{(}{)}{}{}{p}{\varPi}^{\beta}p^{k\beta}\cdot c_{f}(p^{\alpha-2\beta})\sum_{\begin{subarray}{c}{0\leq b<p^{\alpha-\beta}}\\ {(p^{\beta},b)=1}\end{subarray}}\genfrac{(}{)}{}{}{-Nb}{p^{\beta}}^{2k}\mathfrak{e}\left(\frac{b}{p^{\beta}}\right)\psi_{l,\mathbf{r}}(p^{\beta},b).

Remark 9.18.

The sum over $b$ in the above formulas, (81) or (57) is either a partial sum of a geometric sequence or a Gauss sum according to whether $k$ is integral or half-integral. It is not hard to derive an explicit formula for this sum but we exclude this task here. For the $N=1$ case the reader may consult [20, eq. (58)] and [20, Lemma 5.6] for such formulas.

9.4. Identities involving Fourier coefficients of weakly admissible eta-quotients

The main theorem of Section 9 is the following:

Theorem 9.19.

Let $f(\tau)=\prod_{n\mid N}\eta(n\tau)^{r_{n}}=\sum_{n}c_{f}(n)q^{n}$ be a weakly admissible eta-quotient (e.g., a function listed in Table LABEL:table:admissibleTypeI) where $N$ is the level. We keep the notations in (56) with $k^{\prime}=k$ . Then for any $l\in L_{f}$ (c.f. Corollary 9.13) there exists a unique $c_{l}\in\mathbb{C}$ such that

(83)

l^{-\frac{k}{2}}\sum_{\begin{subarray}{c}{a\mid l,\,d=l/a}\\ {(a,N)=1}\end{subarray}}\genfrac{(}{)}{}{}{a}{\varPi}a^{k}\cdot c_{f}\left(\frac{ln}{a^{2}}\right)\sum_{\begin{subarray}{c}{0\leq b<d}\\ {(a,b,d)=1}\end{subarray}}\genfrac{(}{)}{}{}{-Nb}{(a,d)}^{2k}\mathfrak{e}\left(bd\left(\frac{n}{l}-\frac{x_{N}}{24}\right)\right)\psi_{l,\mathbf{r}}(a,b)=c_{l}\cdot c_{f}(n)

holds for any $n\in\frac{x_{N}}{24}+\mathbb{Z}$ . Moreover, the number $c_{l}$ is equal to the left-hand side of the above identity with $n$ replaced by $\frac{x_{N}}{24}$ .

Proof.

By Corollary 9.13 $T_{l}f=c_{l}\cdot f$ and hence $c_{T_{l}f}(n)=c_{l}\cdot c_{f}(n)$ for any $n\in\frac{x_{N}}{24}+\mathbb{Z}$ . Inserting (57) into this identity gives the desired formula. Letting $n=\frac{x_{N}}{24}$ and noting that $c_{f}\left(\frac{x_{N}}{24}\right)=1$ we obtain the formula for $c_{l}$ . ∎

The identities (83) have been verified numerically for $l\leq 121$ , $n-\frac{x_{N}}{24}\leq 40$ and $f$ being any function listed in Table LABEL:table:admissibleTypeI by a SageMath program. See Appendix A for the code. The rest of this section is devoted to useful consequences of Theorem 9.19 and concrete examples.

9.4.1. Closed formulas for Fourier coefficients of eta-quotients.

Let $f(\tau)=\prod_{n\mid N}\eta(n\tau)^{r_{n}}$ be a weakly admissible eta-quotient (e.g., a function listed in Table LABEL:table:admissibleTypeI) where $N$ is the level. We keep the notations in (56) with $k^{\prime}=k$ . Here let us consider the situation $x_{N}=0$ where we have

f(\tau)=\prod_{n\mid N}\prod_{m\in\mathbb{Z}_{\geq 1}}(1-q^{nm})^{r_{n}}=\sum_{n\in\mathbb{Z}_{\geq 0}}c_{f}(n)q^{n},\quad\tau\in\mathfrak{H}.

We have given a series expression of $c_{f}(n)$ involving Dedekind sums for $N=4$ , $r_{1}=2$ , $r_{2}=7$ , $r_{4}=-4$ in Example 8.10. This example is of course also listed in Table LABEL:table:admissibleTypeI. In fact, there exist very simple closed formulas for $c_{f}(n)$ only involving Kronecker-Jacobi symbols at least for $n$ square-free—thanks to Theorem 9.19.

Theorem 9.20.

Let the notations and assumptions be as above. Let $l\in L_{f}$ (c.f. Corollary 9.13) satisfy

(84)

p\nmid N\implies p^{2}\nmid l,\quad\text{for any prime }p.

(e.g. $l$ is square-free or $\mathop{\mathrm{rad}}(l)\mid\mathop{\mathrm{rad}}(N)$ .) We require that at least one of $l$ and $N$ is odd. Then

c_{f}(l)=\begin{dcases}-r_{1}\cdot\sum_{a\mid l,\,(a,N)=1}\genfrac{(}{)}{}{}{a}{\varPi}a^{k-1}&\text{ if }2\mid l,\,2\nmid N,\\ -r_{1}\cdot\sum_{a\mid l,\,(a,N)=1}\genfrac{(}{)}{}{}{a}{\varPi}a^{k-1}\mathfrak{e}\left(\frac{(k+\delta)(l-d)}{4}\right)&\text{ if }2\nmid l,\end{dcases}

where $d=l/a$ , $\varPi=\prod_{n\mid N}(Nn^{-1})^{\lvert r_{n}\rvert}$ and $\delta=0$ ( $\delta=1$ respectively) if $\varPi$ takes the form $2^{\alpha}\cdot(4m+1)$ ( $2^{\alpha}\cdot(4m+3)$ respectively). In particular, if $\mathop{\mathrm{rad}}(l)\mid\mathop{\mathrm{rad}}(N)$ , then $c_{f}(l)=-r_{1}$ .

Proof.

A combination of (82) and (83) (with $n=1$ ) gives $c_{f}(l)\cdot\sum_{0\leq b<l}\psi_{l,\mathbf{r}}(1,b)=l^{\frac{k}{2}}c_{l}\cdot c_{f}(1)$ . By the last assertion of Theorem 9.19 we have

c_{l}=l^{-\frac{k}{2}}\sum_{a\mid l,\,(a,N)=1}\genfrac{(}{)}{}{}{a}{\varPi}a^{k}\cdot c_{f}(0)\sum_{0\leq b<d}\psi_{l,\mathbf{r}}(a,b).

Since $c_{f}(0)=1$ and $c_{f}(1)=-r_{1}$ we find that

(85)

c_{f}(l)\cdot\sum_{0\leq b<l}\psi_{l,\mathbf{r}}(1,b)=-r_{1}\cdot\sum_{a\mid l,\,(a,N)=1}\genfrac{(}{)}{}{}{a}{\varPi}a^{k}\cdot\sum_{0\leq b<d}\psi_{l,\mathbf{r}}(a,b).

The assumption at least one of $l$ and $N$ is odd ensures that $\sum_{0\leq b<l}\psi_{l,\mathbf{r}}(1,b)\neq 0$ . Therefore the desired formulas follow by expanding $\sum_{0\leq b<l}\psi_{l,\mathbf{r}}(1,b)$ and $\sum_{0\leq b<d}\psi_{l,\mathbf{r}}(a,b)$ in (85). ∎

Remark 9.21.

The formulas for $c_{f}(l)$ in the above theorem seem like coefficients of Eisenstein series. This is actually the case since $x_{N}=0$ implies $f$ is not a cusp form. Since $f$ is weakly admissible, then it must be of type I, that is, $\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(N),\chi_{\mathbf{r}})=1$ . Therefore $f$ is in the Eisenstein space.

Remark 9.22.

In the case of half-integral weights, Theorem 9.20 can be used only for those $l$ with $\mathop{\mathrm{rad}}(l)\mid\mathop{\mathrm{rad}}(N)$ . However, there is no half-integral weight eta-quotient $f$ in Table LABEL:table:admissibleTypeI such that $c_{f}(l)$ ( $l>1$ ) can be computed using Theorem 9.20. Nevertheless, it is still worthwhile to include the case of half-integral weights in Theorem 9.20 since there may exist weakly admissible eta-quotients not listed in Table LABEL:table:admissibleTypeI whose coefficients $c_{f}(l)$ can be computed using this theorem.

Example 9.23.

Let us consider an example of level $12$ which is not listed in Table LABEL:table:admissibleTypeI. The reader may use SageMath programs described in Appendix A to produce this eta-quotient and check the assertions given here. Set

f(\tau)=\eta(\tau)^{1}\eta(2\tau)^{-1}\eta(3\tau)^{-1}\eta(4\tau)^{1}\eta(6\tau)^{4}\eta(12\tau)^{-2}=\sum_{n\in\mathbb{Z}_{\geq 0}}c_{f}(n)q^{n}.

Let $(x_{c})_{c\mid 12}$ be as in Proposition 5.1. Hence $\mathop{\mathrm{div}}_{a/c}f=\frac{x_{c}}{24}$ for $c\mid 12$ and $a$ coprime to $c$ ; c.f. (14). We have

\begin{pmatrix}x_{1}\\ x_{2}\\ x_{3}\\ x_{4}\\ x_{6}\\ x_{12}\end{pmatrix}=\begin{pmatrix}12&6&4&3&2&1\\ 3&6&1&3&2&1\\ 4&2&12&1&6&3\\ 3&6&1&12&2&4\\ 1&2&3&1&6&3\\ 1&2&3&4&6&12\end{pmatrix}\begin{pmatrix}r_{1}\\ r_{2}\\ r_{3}\\ r_{4}\\ r_{6}\\ r_{12}\end{pmatrix}=\begin{pmatrix}11\\ 5\\ 9\\ 8\\ 15\\ 0\end{pmatrix},

from which we see that $f$ is a holomorphic eta-quotient. Moreover, $f$ is admissible of type I. Thus Theorem 9.20 can be applied to $f$ . Note that $m_{f}$ in Corollary 9.13 is equal to $24$ and hence

L_{f}=\{l\in\mathbb{Z}_{\geq 1}\colon l\equiv 1\bmod{24}\}.

Therefore $(a,12)=1$ for any $a\mid l$ where $l\in L_{f}$ . Moreover,

k=1,\quad\varPi=2^{9}\cdot 3^{3}=\text{square}\cdot 6,\quad\delta=1.

It follows from Theorem 9.20 that

c_{f}(l)=-\sum_{a\mid l}\genfrac{(}{)}{}{}{a}{6}=-\prod_{p\mid l}\left(1+\genfrac{(}{)}{}{}{p}{6}\right)

where $1\leq l\equiv 1\bmod{24}$ is square-free. Consequently, for such $l$ , $-c_{f}(l)$ is either zero or a power of $2$ . Moreover, for any $\alpha\in\mathbb{Z}_{\geq 1}$ there exist infinitely many $l$ such that $c_{f}(l)=-2^{\alpha}$ .

9.4.2. Multiplicativity of weakly admissible eta-quotients.

The Fourier coefficients $c_{f}(n)$ of an eta-quotient $f$ in Martin’s list [17, Table I] satisfy $c_{f}(nm)c_{f}(1)=c_{f}(n)c_{f}(m)$ whenever $n$ , $m$ are coprime positive integers. This is the reason they are called multiplicative eta-quotients. The proper generalization to weakly admissible eta-quotients is stated in the following theorem.

Theorem 9.24.

Let us keep the notations and assumptions of Theorem 9.19. Then the map $L_{f}\rightarrow\mathbb{C}$ that sends $l$ to $c_{l}$ is a multiplicative function, that is, we have $c_{l_{1}l_{2}}=c_{l_{1}}\cdot c_{l_{2}}$ for any $l_{1},l_{2}\in L_{f}$ with¹⁵¹⁵15The gcd $(l_{1},l_{2})$ is the usual greatest common divisor in the monoid of positive integers, not in the monoid $L_{f}$ . These two concepts are different. For instance in the monoid $L=\{l\in\mathbb{Z}_{\geq 1}\colon l\equiv 1\bmod{24}\}$ , $(5\cdot 29,5\cdot 53)=5$ in the usual sense, but the unique common divisor of $5\cdot 29$ and $5\cdot 53$ in $L$ is $1$ . $(l_{1},l_{2})=1$ .

This theorem is an immediate consequence of the following general fact which in the special case $N=1$ is due to Wohlfahrt [19].

Proposition 9.25.

Let $N\in\mathbb{Z}_{\geq 1}$ and $\chi_{1}$ , $\chi_{2}$ , $\chi_{3}$ be complex linear characters on the double cover $\widetilde{\Gamma_{0}(N)}$ . Let $l_{1},l_{2}\in\mathbb{Z}_{\geq 1}$ such that $\chi_{1}$ and $\chi_{2}$ are $\widetilde{\left(\begin{smallmatrix}{1}&{0}\\ {0}&{l_{1}}\end{smallmatrix}\right)}$ -compatible, $\chi_{2}$ and $\chi_{3}$ are $\widetilde{\left(\begin{smallmatrix}{1}&{0}\\ {0}&{l_{2}}\end{smallmatrix}\right)}$ -compatible. Then $\chi_{1}$ and $\chi_{3}$ are $\widetilde{\left(\begin{smallmatrix}{1}&{0}\\ {0}&{l_{1}l_{2}}\end{smallmatrix}\right)}$ -compatible. Moreover, if $(l_{1},l_{2})=1$ then

(86)

T_{l_{2};\chi_{2},\chi_{3}}\circ T_{l_{1};\chi_{1},\chi_{2}}f=T_{l_{1}l_{2};\chi_{1},\chi_{3}}f

for any meromorphic modular form $f$ on $\Gamma_{0}(N)$ of weight $k\in\frac{1}{2}\mathbb{Z}$ and with character $\chi_{1}$ .

For the concept of compatibility of multiplier systems, see [20, Lemma 3.1] for details. We say $\chi$ and $\chi^{\prime}$ are $\widetilde{\left(\begin{smallmatrix}{1}&{0}\\ {0}&{l}\end{smallmatrix}\right)}$ -compatible if for any $\gamma\in\widetilde{\left(\begin{smallmatrix}{1}&{0}\\ {0}&{l}\end{smallmatrix}\right)}^{-1}\widetilde{\Gamma_{0}(N)}\widetilde{\left(\begin{smallmatrix}{1}&{0}\\ {0}&{l}\end{smallmatrix}\right)}\cap\widetilde{\Gamma_{0}(N)}$ we have $\chi\left(\widetilde{\left(\begin{smallmatrix}{1}&{0}\\ {0}&{l}\end{smallmatrix}\right)}\gamma\widetilde{\left(\begin{smallmatrix}{1}&{0}\\ {0}&{l}\end{smallmatrix}\right)}^{-1}\right)=\chi^{\prime}(\gamma)$ .

Proof of Proposition 9.25 and Theorem 9.24.

First let us prove $\chi_{1}$ and $\chi_{3}$ are $\widetilde{\left(\begin{smallmatrix}{1}&{0}\\ {0}&{l_{1}l_{2}}\end{smallmatrix}\right)}$ -compatible. Let $g\in\widetilde{\left(\begin{smallmatrix}{1}&{0}\\ {0}&{l_{1}l_{2}}\end{smallmatrix}\right)}^{-1}\widetilde{\Gamma_{0}(N)}\widetilde{\left(\begin{smallmatrix}{1}&{0}\\ {0}&{l_{1}l_{2}}\end{smallmatrix}\right)}\cap\widetilde{\Gamma_{0}(N)}$ be arbitrary. We have

	$\displaystyle\chi_{1}\left(\widetilde{\left(\begin{smallmatrix}{1}&{0}\\ {0}&{l_{1}l_{2}}\end{smallmatrix}\right)}g\widetilde{\left(\begin{smallmatrix}{1}&{0}\\ {0}&{l_{1}l_{2}}\end{smallmatrix}\right)}^{-1}\right)$	$\displaystyle=\chi_{1}\left(\widetilde{\left(\begin{smallmatrix}{1}&{0}\\ {0}&{l_{1}}\end{smallmatrix}\right)}\widetilde{\left(\begin{smallmatrix}{1}&{0}\\ {0}&{l_{2}}\end{smallmatrix}\right)}g\widetilde{\left(\begin{smallmatrix}{1}&{0}\\ {0}&{l_{2}}\end{smallmatrix}\right)}^{-1}\widetilde{\left(\begin{smallmatrix}{1}&{0}\\ {0}&{l_{1}}\end{smallmatrix}\right)}^{-1}\right)$
		$\displaystyle=\chi_{2}\left(\widetilde{\left(\begin{smallmatrix}{1}&{0}\\ {0}&{l_{2}}\end{smallmatrix}\right)}g\widetilde{\left(\begin{smallmatrix}{1}&{0}\\ {0}&{l_{2}}\end{smallmatrix}\right)}^{-1}\right)$
		$\displaystyle=\chi_{3}(g).$

In the second equality we have used the assumption $\chi_{1}$ and $\chi_{2}$ are $\widetilde{\left(\begin{smallmatrix}{1}&{0}\\ {0}&{l_{1}}\end{smallmatrix}\right)}$ -compatible and in the third equality the assumption $\chi_{2}$ and $\chi_{3}$ are $\widetilde{\left(\begin{smallmatrix}{1}&{0}\\ {0}&{l_{2}}\end{smallmatrix}\right)}$ -compatible. This proves the $\widetilde{\left(\begin{smallmatrix}{1}&{0}\\ {0}&{l_{1}l_{2}}\end{smallmatrix}\right)}$ -compatibility of $\chi_{1}$ and $\chi_{3}$ .

Assume that $(l_{1},l_{2})=1$ . To prove (86) we need an equivalent definition of $T_{l}f$ via double coset actions. For the proof of this equivalence see [20, eq. (28) and (30)]. Write

\widetilde{\Gamma_{0}(N)}\widetilde{\left(\begin{smallmatrix}{1}&{0}\\ {0}&{l_{i}}\end{smallmatrix}\right)}\widetilde{\Gamma_{0}(N)}=\bigsqcup_{\gamma\in A_{i}}\widetilde{\Gamma_{0}(N)}\widetilde{\gamma}

for $i=1,2$ . Then

T_{l_{1};\chi_{1},\chi_{2}}f=\sum_{\gamma\in A_{1}}(\chi_{1}|_{l_{1}}\chi_{2})^{-1}(\widetilde{\gamma})\cdot f|_{k}\widetilde{\gamma},

where $\chi_{1}|_{l_{1}}\chi_{2}(h)=\chi_{1}(h_{1})\chi_{2}(h_{2})$ if we write $h=h_{1}\widetilde{\left(\begin{smallmatrix}{1}&{0}\\ {0}&{l_{1}}\end{smallmatrix}\right)}h_{2}$ with $h_{1},h_{2}\in\widetilde{\Gamma_{0}(N)}$ . The compatibility ensures that $\chi_{1}|_{l_{1}}\chi_{2}(h)$ is well-defined. Therefore

(87)

T_{l_{2};\chi_{2},\chi_{3}}\circ T_{l_{1};\chi_{1},\chi_{2}}f=\sum_{\gamma_{1}\in A_{1},\gamma_{2}\in A_{2}}(\chi_{2}|_{l_{2}}\chi_{3})^{-1}(\widetilde{\gamma_{2}})(\chi_{1}|_{l_{1}}\chi_{2})^{-1}(\widetilde{\gamma_{1}})\cdot f|_{k}\widetilde{\gamma_{1}}\widetilde{\gamma_{2}}.

By [20, eq. (25)] with $m=1$ we can choose the following $A_{i}$ :

(88)

A_{i}=\left\{\left(\begin{smallmatrix}{a}&{b}\\ {0}&{d}\end{smallmatrix}\right)\colon a,d>0,\,0\leq b<d,\,ad=l_{i},\,(N,a)=1,\,(a,b,d)=1\right\}.

It follows from the assumption $(l_{1},l_{2})=1$ and elementary number theory that $\widetilde{\gamma_{1}}\widetilde{\gamma_{2}}\in\widetilde{\Gamma_{0}(N)}\widetilde{\left(\begin{smallmatrix}{1}&{0}\\ {0}&{l_{1}l_{2}}\end{smallmatrix}\right)}\widetilde{\Gamma_{0}(N)}$ if $\gamma_{1}\in A_{1}$ and $\gamma_{2}\in A_{2}$ . Therefore the map

	$\displaystyle\sigma\colon A_{1}\times A_{2}$	$\displaystyle\rightarrow\widetilde{\Gamma_{0}(N)}\backslash\widetilde{\Gamma_{0}(N)}\widetilde{\left(\begin{smallmatrix}{1}&{0}\\ {0}&{l_{1}l_{2}}\end{smallmatrix}\right)}\widetilde{\Gamma_{0}(N)}$
	$\displaystyle(\gamma_{1},\gamma_{2})$	$\displaystyle\mapsto\widetilde{\Gamma_{0}(N)}\cdot\widetilde{\gamma_{1}}\widetilde{\gamma_{2}}$

is well-defined. By the theory of classical Hecke algebras, or more precisely, by Theorem 2.8.1(2) and Eq. (2.7.2) of [35], $\sigma$ is a surjection and each inverse image $\sigma^{-1}(x)$ has the same cardinality. Since $\sigma^{-1}\left(\widetilde{\Gamma_{0}(N)}\widetilde{\left(\begin{smallmatrix}{1}&{0}\\ {0}&{l_{1}l_{2}}\end{smallmatrix}\right)}\right)=\left\{(\left(\begin{smallmatrix}{1}&{0}\\ {0}&{l_{1}}\end{smallmatrix}\right),\left(\begin{smallmatrix}{1}&{0}\\ {0}&{l_{2}}\end{smallmatrix}\right))\right\}$ is a singleton, $\sigma$ is a bijection which means the range of summation in the right-hand side of (87) is essentially a set of representatives of the coset space $\widetilde{\Gamma_{0}(N)}\backslash\widetilde{\Gamma_{0}(N)}\widetilde{\left(\begin{smallmatrix}{1}&{0}\\ {0}&{l_{1}l_{2}}\end{smallmatrix}\right)}\widetilde{\Gamma_{0}(N)}$ . Now we can define a map, which is key to the proof,

	$\displaystyle\xi\colon\widetilde{\Gamma_{0}(N)}\widetilde{\left(\begin{smallmatrix}{1}&{0}\\ {0}&{l_{1}l_{2}}\end{smallmatrix}\right)}\widetilde{\Gamma_{0}(N)}$	$\displaystyle\rightarrow\mathbb{C}$
	$\displaystyle x$	$\displaystyle\mapsto\chi_{1}\|_{l_{1}}\chi_{2}(x\widetilde{\gamma}_{2}^{-1})\cdot\chi_{2}\|_{l_{2}}\chi_{3}(\widetilde{\gamma}_{2})$

where $(\gamma_{1},\gamma_{2})$ is the uniquely determined pair in $A_{1}\times A_{2}$ such that $\widetilde{\Gamma_{0}(N)}x=\widetilde{\Gamma_{0}(N)}\widetilde{\gamma_{1}}\widetilde{\gamma_{2}}$ . It is not hard to see that

(89)

\xi(gxh)=\chi_{1}(g)\xi(x)\chi_{3}(h),\quad g,h\in\widetilde{\Gamma_{0}(N)},\,x\in\widetilde{\Gamma_{0}(N)}\widetilde{\left(\begin{smallmatrix}{1}&{0}\\ {0}&{l_{1}l_{2}}\end{smallmatrix}\right)}\widetilde{\Gamma_{0}(N)},\quad\xi\widetilde{\left(\begin{smallmatrix}{1}&{0}\\ {0}&{l_{1}l_{2}}\end{smallmatrix}\right)}=1

and that

(90)

\xi(\widetilde{\gamma_{1}}\widetilde{\gamma_{2}})=\chi_{1}|_{l_{1}}\chi_{2}(\widetilde{\gamma_{1}})\chi_{2}|_{l_{2}}\chi_{3}(\widetilde{\gamma_{2}}).

Since the map $x\mapsto\chi_{1}|_{l_{1}l_{2}}\chi_{3}(x)$ also satisfies (89), we have $\xi=\chi_{1}|_{l_{1}l_{2}}\chi_{3}$ . Inserting this and (90) into (87) gives

T_{l_{2};\chi_{2},\chi_{3}}\circ T_{l_{1};\chi_{1},\chi_{2}}f=\sum_{\gamma_{1}\in A_{1},\gamma_{2}\in A_{2}}\xi^{-1}(\widetilde{\gamma_{1}}\widetilde{\gamma_{2}})\cdot f|_{k}\widetilde{\gamma_{1}}\widetilde{\gamma_{2}}=\sum_{\gamma_{1}\in A_{1},\gamma_{2}\in A_{2}}(\chi_{1}|_{l_{1}l_{2}}\chi_{3})^{-1}(\widetilde{\gamma_{1}}\widetilde{\gamma_{2}})\cdot f|_{k}\widetilde{\gamma_{1}}\widetilde{\gamma_{2}}.

Since $\sigma$ is a bijection, we obtain (86) from the above equalities.

Finally we prove Theorem 9.24. Let $f$ be as in this theorem. Then $T_{l_{1}}f=c_{l_{1}}\cdot f$ , $T_{l_{2}}f=c_{l_{2}}\cdot f$ and $T_{l_{1}l_{2}}f=c_{l_{1}l_{2}}\cdot f$ by Corollary 9.13. Since $(l_{1},l_{2})=1$ we have

c_{l_{1}l_{2}}\cdot f=T_{l_{1}l_{2}}f=T_{l_{2}}(T_{l_{1}}f)=T_{l_{2}}(c_{l_{1}}\cdot f)=c_{l_{1}}c_{l_{2}}\cdot f

where we have used Proposition 9.25 in the second equality. Since $f$ is not identically zero we find $c_{l_{1}l_{2}}=c_{l_{1}}\cdot c_{l_{2}}$ as required. ∎

Before looking at examples, let us simplify some notations. Set $x_{N}^{\prime}=\frac{x_{N}}{24}$ and

S_{l,\mathbf{r}}(a)=\sum_{\begin{subarray}{c}{0\leq b<d}\\ {(a,b,d)=1}\end{subarray}}\genfrac{(}{)}{}{}{-Nb}{(a,d)}^{2k}\mathfrak{e}\left(-\frac{(l-1)x_{N}^{\prime}b}{a}\right)\psi_{l,\mathbf{r}}(a,b).

Then according to Theorem 9.19 we have

c_{l}=l^{-\frac{k}{2}}\sum_{a\mid l,\,(a,N)=1}\genfrac{(}{)}{}{}{a}{\varPi}a^{k}\cdot c_{f}\left(\frac{lx_{N}^{\prime}}{a^{2}}\right)S_{l,\mathbf{r}}(a),\quad l\in L_{f}.

We recall that here and below $f$ must be a weakly admissible eta-quotient.

Example 9.26.

If $x_{N}^{\prime}=0$ , that is, $x_{N}=\sum_{n\mid N}nr_{n}=0$ , then $c_{f}(0)=1$ and hence Theorem 9.24 gives

\sum_{a\mid l_{1},\,(a,N)=1}\genfrac{(}{)}{}{}{a}{\varPi}a^{k}\cdot S_{l_{1},\mathbf{r}}(a)\cdot\sum_{a\mid l_{2},\,(a,N)=1}\genfrac{(}{)}{}{}{a}{\varPi}a^{k}\cdot S_{l_{2},\mathbf{r}}(a)=\sum_{a\mid l_{1}l_{2},\,(a,N)=1}\genfrac{(}{)}{}{}{a}{\varPi}a^{k}\cdot S_{l_{1}l_{2},\mathbf{r}}(a)

for $l_{1},l_{2}\in L_{f}$ with $(l_{1},l_{2})=1$ .

Example 9.27.

If $x_{N}^{\prime}\in\mathbb{Z}_{\geq 0}$ and $(l,x_{N}^{\prime})=1$ , then $c_{f}\left(\frac{lx_{N}^{\prime}}{a^{2}}\right)=0$ unless $\frac{lx_{N}^{\prime}}{a^{2}}\in\mathbb{Z}$ by Theorem 9.15(a). Therefore we have

\sum_{a^{2}\mid l_{1},\,(a,N)=1}\genfrac{(}{)}{}{}{a}{\varPi}a^{k}\cdot c_{f}\left(\frac{l_{1}x_{N}^{\prime}}{a^{2}}\right)S_{l_{1},\mathbf{r}}(a)\cdot\sum_{a^{2}\mid l_{2},\,(a,N)=1}\genfrac{(}{)}{}{}{a}{\varPi}a^{k}\cdot c_{f}\left(\frac{l_{2}x_{N}^{\prime}}{a^{2}}\right)S_{l_{2},\mathbf{r}}(a)\\ =\sum_{a^{2}\mid l_{1}l_{2},\,(a,N)=1}\genfrac{(}{)}{}{}{a}{\varPi}a^{k}\cdot c_{f}\left(\frac{l_{1}l_{2}x_{N}^{\prime}}{a^{2}}\right)S_{l_{1}l_{2},\mathbf{r}}(a)

if $l_{1},l_{2}\in L_{f}$ with $(l_{1},l_{2})=1$ , $(l_{1},x_{N}^{\prime})=1$ and $(l_{2},x_{N}^{\prime})=1$ . If in addition (84) holds for $l=l_{1},l_{2}$ , then the above identity becomes

(91)

c_{f}(l_{1}x_{N}^{\prime})S_{l_{1},\mathbf{r}}(1)\cdot c_{f}(l_{2}x_{N}^{\prime})S_{l_{2},\mathbf{r}}(1)=c_{f}(l_{1}l_{2}x_{N}^{\prime})S_{l_{1}l_{2},\mathbf{r}}(1).

From this we obtain a property of $c_{f}(n)$ very closed to the multiplicativity property. (We exclude the half-integral case below for simplicity.)

Proposition 9.28.

Let $f(\tau)=\prod_{n\mid N}\eta(n\tau)^{r_{n}}=\sum_{n}c_{f}(n)q^{n}$ be a weakly admissible eta-quotient (e.g., a function listed in Table LABEL:table:admissibleTypeI or Example 9.23) where $N$ is the level. We keep the notations in (56) with $k^{\prime}=k$ . Assume that $x_{N}^{\prime}=\frac{x_{N}}{24}\in\mathbb{Z}_{\geq 0}$ and $k\in\mathbb{Z}$ . Let $l_{1},l_{2}\in L_{f}$ satisfy $(l_{1},l_{2})=1$ , $(l_{1},x_{N}^{\prime})=1$ and $(l_{2},x_{N}^{\prime})=1$ . We require (84) holds for $l=l_{1}$ and $l_{2}$ . Moreover, if $N\equiv 0\bmod{2}$ and $N(k+\delta)\equiv 2\bmod{4}$ , then we require $l_{1}\equiv l_{2}\equiv 1\bmod{2}$ . (See Theorem 9.15(b) for the definition of $\delta$ .) Then

c_{f}(l_{1}x_{N}^{\prime})\cdot c_{f}(l_{2}x_{N}^{\prime})=\varepsilon\cdot c_{f}(l_{1}l_{2}x_{N}^{\prime}),

where $\varepsilon=u_{l_{1},\mathbf{r}}\cdot u_{l_{2},\mathbf{r}}\cdot u_{l_{1}l_{2},\mathbf{r}}$ with

u_{l,\mathbf{r}}=\begin{dcases}1,&\text{ if }2\mid l,\quad 2\nmid N\text{ or }4\mid N(k+\delta)\\ 1,&\text{ if }2\nmid l,\quad 4\mid N(k+\delta)(l-1)\\ -1,&\text{ if }2\nmid l,\quad 4\nmid N(k+\delta)(l-1).\end{dcases}

Proof.

According to (91) we need to prove

(92)

S_{l_{1}l_{2},\mathbf{r}}(1)=u_{l_{1},\mathbf{r}}\cdot u_{l_{2},\mathbf{r}}\cdot u_{l_{1}l_{2},\mathbf{r}}\cdot S_{l_{1},\mathbf{r}}(1)\cdot S_{l_{2},\mathbf{r}}(1).

By the definition of $\psi_{l,\mathbf{r}}$ in Theorem 9.15, we have, since $k\in\mathbb{Z}$ ,

S_{l,\mathbf{r}}(1)=\begin{dcases}l,&\text{ if }2\mid l,\quad 2\nmid N\text{ or }4\mid N(k+\delta)\\ 0,&\text{ if }2\mid l,\quad 2\mid N\text{ and }N(k+\delta)\equiv 2\bmod{4}\\ l,&\text{ if }2\nmid l,\quad 4\mid N(k+\delta)(l-1)\\ -l,&\text{ if }2\nmid l,\quad 4\nmid N(k+\delta)(l-1).\end{dcases}

From this (92) follows. Substituting (92) in (91) we obtain

c_{f}(l_{1}x_{N}^{\prime})\cdot c_{f}(l_{2}x_{N}^{\prime})S_{l_{1},\mathbf{r}}(1)S_{l_{2},\mathbf{r}}(1)=\varepsilon\cdot c_{f}(l_{1}l_{2}x_{N}^{\prime})S_{l_{1},\mathbf{r}}(1)S_{l_{2},\mathbf{r}}(1).

By the assumption we have $S_{l_{1},\mathbf{r}}(1)$ and $S_{l_{2},\mathbf{r}}(1)$ are nonzero and hence the required relation follows. ∎

If $x_{N}^{\prime}=0$ , then the above proposition holds trivially. We believe there are many weakly admissible eta-quotients with $x_{N}^{\prime}\in\mathbb{Z}_{\geq 1}$ to which we can apply the above proposition. The most well-known example is the discriminant function $\Delta(\tau)=\eta(\tau)^{24}$ , which in our terminology is an admissible eta-quotient of type II. The only example in Table LABEL:table:admissibleTypeI is the one presented below.

Example 9.29.

Set $f=\eta(3\tau)^{2}\eta(9\tau)^{-1}\eta(27\tau)=\sum_{n\in\mathbb{Z}_{\geq 1}}c_{f}(n)q^{n}$ . This is an admissible eta-quotient of type I. We have

N=27,\quad k=1,\quad x_{N}=24,\quad\varPi=3^{5},\quad\delta=1.

Since (c.f. (19))

\begin{pmatrix}x_{1}\\ x_{3}\\ x_{9}\\ x_{27}\end{pmatrix}=\begin{pmatrix}27&9&3&1\\ 3&9&3&1\\ 1&3&9&3\\ 1&3&9&27\end{pmatrix}\begin{pmatrix}r_{1}\\ r_{3}\\ r_{9}\\ r_{27}\end{pmatrix}=\begin{pmatrix}16\\ 16\\ 0\\ 24\end{pmatrix},

we have $m_{f}=3$ (c.f. Corollary 9.13) and hence $L_{f}=\{3n+1\colon n\in\mathbb{Z}_{\geq 0}\}$ . It follows from Proposition 9.28 that $c_{f}(l_{1})c_{f}(l_{2})=c_{f}(l_{1}l_{2})$ where $l_{1},l_{2}\equiv 1\bmod{3}$ are square-free positive integers with $(l_{1},l_{2})=1$ .

Example 9.30.

Let us give an example of admissible eta-quotient of type II. Set

f=\eta(\tau)\eta(2\tau)\eta(3\tau)\eta(6\tau)^{3}=\sum_{n\in\mathbb{Z}_{\geq 1}}c_{f}(n)q^{n}.

We have

N=6,\quad k=3,\quad x_{N}=24,\quad\varPi=2^{3}\cdot 3^{2},\quad\delta=0.

Since

\begin{pmatrix}x_{1}\\ x_{2}\\ x_{3}\\ x_{6}\end{pmatrix}=\begin{pmatrix}6&3&2&1\\ 3&6&1&2\\ 2&1&6&3\\ 1&2&3&6\end{pmatrix}\begin{pmatrix}r_{1}\\ r_{2}\\ r_{3}\\ r_{6}\end{pmatrix}=\begin{pmatrix}14\\ 16\\ 18\\ 24\end{pmatrix},

we have $m_{f}=12$ and hence $L_{f}=\{12n+1\colon n\in\mathbb{Z}_{\geq 0}\}$ . By Theorem 4.2, $\dim_{\mathbb{C}}M_{3}(\Gamma_{0}(6),\chi)=2$ where $\chi\colon\widetilde{\Gamma_{0}(6)}\rightarrow\mathbb{C}^{\times}$ is the character of $f$ . (Of course this descends to a character of the matrix group $\Gamma_{0}(6)$ since $k\in\mathbb{Z}$ .) Recall that the number of Eisenstein series in $M_{3}(\Gamma_{0}(6),\chi)$ is denoted by $n_{0}$ in (50). Since $\chi(\widetilde{\gamma_{a/c}}\widetilde{T}^{w_{a/c}}\widetilde{\gamma_{a/c}}^{-1})=1$ if and only if $x_{c}\equiv 0\bmod{24}$ where $c\mid 6$ and $(a,c)=1$ , among the four cusps in $\Gamma_{0}(6)\backslash\mathbb{P}^{1}(\mathbb{Q})$ there is only one Eisenstein series which is defined at the cusp $\mathrm{i}\infty$ . Therefore $n_{0}=1$ . Since $f$ is a cusp form, we find that it is admissible of type II. It follows from Proposition 9.28 (applied to this $f$ ) that $c_{f}(l_{1})c_{f}(l_{2})=c_{f}(l_{1}l_{2})$ whenever $l_{1},l_{2}\equiv 1\bmod{12}$ are square-free positive integers with $(l_{1},l_{2})=1$ . It is worthwhile to notice that for other pair $(l_{1},l_{2})$ , the multiplicativity does not necessarily hold. For instance, $c_{f}(2)=-1$ , $c_{f}(3)=-2$ but $c_{f}(6)=4$ .

9.4.3. An example of half-integral weight

The examples presented above are all of integral weights. To conclude this section we supplement an example of half-integral weight where we show how the Gauss sum $\sum_{b}$ in (81) can be simplified. Let

(93)

f(\tau)=\eta(\tau)^{-7}\eta(2\tau)^{17}\eta(4\tau)^{-3}=\sum_{n\in\frac{5}{8}+\mathbb{Z}_{\geq 0}}c_{f}(n)q^{n}.

The statistics of $f$ are

N=4,\quad k=\frac{7}{2},\quad x_{1}=3,\quad,x_{2}=24,\quad x_{4}=15,\quad\varPi=2^{31},\quad\delta=0,\quad m_{f}=8.

Thus $L_{f}=\{l^{2}\colon l\equiv 1\bmod{2}\}$ . By Theorem 4.2 we have $\dim_{\mathbb{C}}M_{7/2}(\Gamma_{0}(4),\chi)=2$ where $\chi\colon\widetilde{\Gamma_{0}(4)}\rightarrow\mathbb{C}^{\times}$ is the character of $f$ and by Definition 8.3 the Eisenstein series can only be given at the cusp $1/2$ . Thus, $f$ is admissible of type II and Theorems 9.19, 9.24 can be applied.

Now we calculate $c_{l^{2}}$ for $l^{2}\in L_{f}$ (c.f. Theorem 9.19). We have, by definition,

(94)

c_{l^{2}}=l^{-\frac{7}{2}}\sum_{a\mid l^{2}}\genfrac{(}{)}{}{}{a}{2}a^{\frac{7}{2}}c_{f}\left(\frac{5l^{2}}{8a^{2}}\right)\mathfrak{e}\left(-\frac{7l^{2}}{8a}+\frac{7}{8}\right)\sum_{\begin{subarray}{c}{0\leq b<l^{2}/a}\\ {(a,b,l^{2}/a)=1}\end{subarray}}\genfrac{(}{)}{}{}{-b}{(a,l^{2}/a)}\mathfrak{e}\left(-\frac{5(l^{2}-1)b}{8a}\right).

Note that $c_{f}\left(\frac{5l^{2}}{8a^{2}}\right)=0$ unless $\frac{5l^{2}}{8a^{2}}\in\frac{5}{8}+\mathbb{Z}$ , that is, unless $a\mid l$ since $a,\,l$ are odd. It follows that for such $a$ we have $(a,l^{2}/a)=a$ ,

t:=\frac{5(l^{2}-1)\cdot(a,l^{2}/a)}{8a}=\frac{5(l^{2}-1)}{8}

is an integer (depending on $a$ and $l$ ) and

	$\displaystyle\sum_{\begin{subarray}{c}{0\leq b<l^{2}/a}\\ {(a,b,l^{2}/a)=1}\end{subarray}}\genfrac{(}{)}{}{}{-b}{(a,l^{2}/a)}\mathfrak{e}\left(-\frac{5(l^{2}-1)b}{8a}\right)$	$\displaystyle=\sum_{0\leq b<l^{2}/a}\genfrac{(}{)}{}{}{-b}{a}\mathfrak{e}\left(\frac{-tb}{a}\right)$
(95)			$\displaystyle=\frac{l^{2}}{a^{2}}\sum_{0\leq b<a}\genfrac{(}{)}{}{}{b}{a}\mathfrak{e}\left(\frac{tb}{a}\right)$

is, up to a simple factor, a Gauss sum associated with the character $\genfrac{(}{)}{}{}{\cdot}{(a,l^{2}/a)}$ . To express the values of these Gauss sums, we let, for any odd positive integer $m$ ,

$\displaystyle\mathop{\mathrm{rad}_{E}}(m)$	$\displaystyle=\prod_{\begin{subarray}{c}{p\mid m}\\ {v_{p}(m)\equiv 0\bmod{2}}\end{subarray}}p\qquad$	$\displaystyle\mathop{\mathrm{rad}_{O}}(m)$	$\displaystyle=\prod_{\begin{subarray}{c}{p\mid m}\\ {v_{p}(m)\equiv 1\bmod{2}}\end{subarray}}p$
$\displaystyle\mathop{\mathrm{rad}}(m)$	$\displaystyle=\mathop{\mathrm{rad}_{E}}(m)\mathop{\mathrm{rad}_{O}}(m)\qquad$	$\displaystyle\mathop{\mathrm{rad}^{\prime}}(m)$	$\displaystyle=\mathop{\mathrm{rad}_{E}}(m)^{2}\mathop{\mathrm{rad}_{O}}(m)$
$\displaystyle\mathop{\mathrm{irad}}(m)$	$\displaystyle=m/\mathop{\mathrm{rad}}(m)\qquad$	$\displaystyle\mathop{\mathrm{irad}^{\prime}}(m)$	$\displaystyle=m/\mathop{\mathrm{rad}^{\prime}}(m)$

where $p$ denotes a prime. By [20, Lemma 5.6], if $\mathop{\mathrm{irad}}(a)\nmid t$ then $\sum_{0\leq b<a}\genfrac{(}{)}{}{}{b}{a}\mathfrak{e}\left(\frac{tb}{a}\right)=0$ ; if $\mathop{\mathrm{irad}}(a)\mid t$ then

(96)

\sum_{0\leq b<a}\genfrac{(}{)}{}{}{b}{a}\mathfrak{e}\left(\frac{tb}{a}\right)=\varepsilon_{a}\frac{a}{\sqrt{\mathop{\mathrm{rad}^{\prime}}(a)}}\genfrac{(}{)}{}{}{t/\mathop{\mathrm{irad}^{\prime}}(a)}{\mathop{\mathrm{rad}_{O}}(a)}\prod_{p\mid\mathop{\mathrm{rad}_{E}}(a)}\left(p-1-p\genfrac{(}{)}{}{}{t/\mathop{\mathrm{irad}}(a)}{p}^{2}\right),

where $\genfrac{(}{)}{}{}{t/\mathop{\mathrm{irad}^{\prime}}(a)}{\mathop{\mathrm{rad}_{O}}(a)}$ , $\genfrac{(}{)}{}{}{t/\mathop{\mathrm{irad}}(a)}{p}$ are Jacobi symbols and $\varepsilon_{a}=1$ ( $\varepsilon_{a}=\mathrm{i}$ respectively) if $a\equiv 1\bmod{4}$ ( $a\equiv 3\bmod{4}$ respectively). Inserting (96) and (95) into (94) and noting that $\mathfrak{e}\left(-\frac{7l^{2}}{8a}+\frac{7}{8}\right)=\mathfrak{e}\left(\frac{a-1}{8}\right)$ , $\genfrac{(}{)}{}{}{a}{2}\varepsilon_{a}\mathfrak{e}\left(\frac{a-1}{8}\right)=1$ we obtain the following formula.

Proposition 9.31.

For $f(\tau)=\eta(\tau)^{-7}\eta(2\tau)^{17}\eta(4\tau)^{-3}=\sum_{n\in\frac{5}{8}+\mathbb{Z}_{\geq 0}}c_{f}(n)q^{n}$ and $1\leq l\equiv 1\bmod{2}$ we have

c_{l^{2}}=l^{-\frac{3}{2}}\sum_{\begin{subarray}{c}{a\mid l}\\ {\mathop{\mathrm{irad}}(a)\mid t}\end{subarray}}\frac{a^{\frac{5}{2}}}{\sqrt{\mathop{\mathrm{rad}^{\prime}}(a)}}\prod_{p\mid\mathop{\mathrm{rad}_{E}}(a)}\left(p-1-p\genfrac{(}{)}{}{}{t/\mathop{\mathrm{irad}}(a)}{p}^{2}\right)\genfrac{(}{)}{}{}{t/\mathop{\mathrm{irad}^{\prime}}(a)}{\mathop{\mathrm{rad}_{O}}(a)}c_{f}\left(\frac{5l^{2}}{8a^{2}}\right)

where $t=\frac{5(l^{2}-1)}{8}$ . In particular, if $l$ is square-free, then

c_{l^{2}}=l^{-\frac{3}{2}}\sum_{a\mid l}a^{2}\cdot\genfrac{(}{)}{}{}{t}{a}c_{f}\left(\frac{5l^{2}}{8a^{2}}\right).

Remark 9.32.

Since $\frac{a^{5/2}}{\sqrt{\mathop{\mathrm{rad}^{\prime}}(a)}}\in\mathbb{Z}$ and $c_{f}(n)\in\mathbb{Z}$ , we have $l^{3/2}\cdot c_{l^{2}}\in\mathbb{Z}$ .

According to Theorem 9.24 (applied to (93)) we have $c_{l_{1}^{2}}c_{l_{2}^{2}}=c_{l_{1}^{2}l_{2}^{2}}$ whenever $l_{1},l_{2}$ are coprime odd integers. In the special case $l_{1}$ and $l_{2}$ are square-free, this identity reads

(97)

\sum_{a\mid l_{1}}a^{2}\cdot\genfrac{(}{)}{}{}{5(l_{1}^{2}-1)/8}{a}c_{f}\left(\frac{5l_{1}^{2}}{8a^{2}}\right)\cdot\sum_{a\mid l_{2}}a^{2}\cdot\genfrac{(}{)}{}{}{5(l_{2}^{2}-1)/8}{a}c_{f}\left(\frac{5l_{2}^{2}}{8a^{2}}\right)\\ =\sum_{a\mid l_{1}l_{2}}a^{2}\cdot\genfrac{(}{)}{}{}{5(l_{1}^{2}l_{2}^{2}-1)/8}{a}c_{f}\left(\frac{5l_{1}^{2}l_{2}^{2}}{8a^{2}}\right).

From this we find the genuine multiplicativity property of (93) as stated below. The reader may compare this to the integral-weight counterpart—Proposition 9.28.

Proposition 9.33.

Let $f(\tau)=\eta(\tau)^{-7}\eta(2\tau)^{17}\eta(4\tau)^{-3}=\sum_{n\in\frac{5}{8}+\mathbb{Z}_{\geq 0}}c_{f}(n)q^{n}$ . Let $l_{1},l_{2}$ be coprime odd square-free integers. Then

c_{f}\left(\frac{5l_{1}^{2}}{8}\right)c_{f}\left(\frac{5l_{2}^{2}}{8}\right)=c_{f}\left(\frac{5l_{1}^{2}l_{2}^{2}}{8}\right).

Proof.

First suppose $l_{1}$ and $l_{2}$ are primes. By (97) we have

(98)

\left(c_{f}\left(\frac{5l_{1}^{2}}{8}\right)+l_{1}^{2}\cdot\genfrac{(}{)}{}{}{5(l_{1}^{2}-1)/8}{l_{1}}\right)\cdot\left(c_{f}\left(\frac{5l_{2}^{2}}{8}\right)+l_{2}^{2}\cdot\genfrac{(}{)}{}{}{5(l_{2}^{2}-1)/8}{l_{2}}\right)\\ =c_{f}\left(\frac{5l_{1}^{2}l_{2}^{2}}{8}\right)+l_{2}^{2}\cdot\genfrac{(}{)}{}{}{5(l_{1}^{2}l_{2}^{2}-1)/8}{l_{2}}c_{f}\left(\frac{5l_{1}^{2}}{8}\right)+l_{1}^{2}\cdot\genfrac{(}{)}{}{}{5(l_{1}^{2}l_{2}^{2}-1)/8}{l_{1}}c_{f}\left(\frac{5l_{2}^{2}}{8}\right)+l_{1}^{2}l_{2}^{2}\cdot\genfrac{(}{)}{}{}{5(l_{1}^{2}l_{2}^{2}-1)/8}{l_{1}l_{2}}.

It is not hard to prove, by the periodicity and multiplicativity of Jacobi symbols, that

	$\displaystyle\genfrac{(}{)}{}{}{5(l_{1}^{2}l_{2}^{2}-1)/8}{l_{1}}$	$\displaystyle=\genfrac{(}{)}{}{}{5(l_{1}^{2}-1)/8}{l_{1}},$
	$\displaystyle\genfrac{(}{)}{}{}{5(l_{1}^{2}l_{2}^{2}-1)/8}{l_{2}}$	$\displaystyle=\genfrac{(}{)}{}{}{5(l_{2}^{2}-1)/8}{l_{2}},$
	$\displaystyle\genfrac{(}{)}{}{}{5(l_{1}^{2}l_{2}^{2}-1)/8}{l_{1}l_{2}}$	$\displaystyle=\genfrac{(}{)}{}{}{5(l_{1}^{2}-1)/8}{l_{1}}\cdot\genfrac{(}{)}{}{}{5(l_{2}^{2}-1)/8}{l_{2}}.$

Inserting these into (98) gives the desired identity in the case $l_{1}$ and $l_{2}$ are primes.

Second, suppose $l_{1}$ is a prime and $l_{2}$ is arbitrary. Again by (97) we have

(99)

\left(c_{f}\left(\frac{5l_{1}^{2}}{8}\right)+l_{1}^{2}\cdot\genfrac{(}{)}{}{}{5(l_{1}^{2}-1)/8}{l_{1}}\right)\cdot\sum_{a\mid l_{2}}a^{2}\cdot\genfrac{(}{)}{}{}{5(l_{2}^{2}-1)/8}{a}c_{f}\left(\frac{5l_{2}^{2}}{8a^{2}}\right)\\ =\sum_{a\mid l_{2}}a^{2}\cdot\genfrac{(}{)}{}{}{5(l_{1}^{2}l_{2}^{2}-1)/8}{a}c_{f}\left(\frac{5l_{1}^{2}l_{2}^{2}}{8a^{2}}\right)+\sum_{a\mid l_{2}}l_{1}^{2}a^{2}\cdot\genfrac{(}{)}{}{}{5(l_{1}^{2}l_{2}^{2}-1)/8}{l_{1}a}c_{f}\left(\frac{5l_{2}^{2}}{8a^{2}}\right).

Since for $a\mid l_{2}$ ,

\genfrac{(}{)}{}{}{5(l_{1}^{2}-1)/8}{l_{1}}\cdot\genfrac{(}{)}{}{}{5(l_{2}^{2}-1)/8}{a}=\genfrac{(}{)}{}{}{5(l_{1}^{2}l_{2}^{2}-1)/8}{l_{1}a},

we obtain from (99) that

	$\displaystyle c_{f}\left(\frac{5l_{1}^{2}}{8}\right)\cdot\sum_{a\mid l_{2}}a^{2}\cdot\genfrac{(}{)}{}{}{5(l_{2}^{2}-1)/8}{a}c_{f}\left(\frac{5l_{2}^{2}}{8a^{2}}\right)$	$\displaystyle=\sum_{a\mid l_{2}}a^{2}\cdot\genfrac{(}{)}{}{}{5(l_{1}^{2}l_{2}^{2}-1)/8}{a}c_{f}\left(\frac{5l_{1}^{2}l_{2}^{2}}{8a^{2}}\right)$
(100)			$\displaystyle=\sum_{a\mid l_{2}}a^{2}\cdot\genfrac{(}{)}{}{}{5(l_{2}^{2}-1)/8}{a}c_{f}\left(\frac{5l_{1}^{2}l_{2}^{2}}{8a^{2}}\right).$

The desired identities for $l_{1}$ prime and $l_{2}$ arbitrary now follows from (100), the already proven case and the induction on the number of prime divisors of $l_{2}$ .

Finally, if $l_{1}$ and $l_{2}$ are arbitrary, then, by the already proven cases, both sides of the desired identity are equal to $\prod_{p\mid l_{1}l_{2}}c_{f}\left(\frac{5p^{2}}{8}\right)$ and hence the identity holds which concludes the proof. ∎

We have discussed consequences of Theorem 9.24 applied to (93). The above exploration is also valid for any admissible eta-quotient of half-integral weight. (Of course the integral weight case is simpler to deal with.) In addition, for (93) or any admissible eta-quotient of half-integral weight, we can as well simplify the left-hand side of (83) as we have just done for $c_{l^{2}}$ . This will give more interesting identities about $c_{f}(n)$ . We will not include more details about this due to the length of the paper.

10. Miscellaneous observations and open questions

More general multiplier systems. The dimension formulas stated in Theorem 4.2 concern multiplier systems induced by eta-quotients of fractional exponents. As a comparison, most of the formulas that have appeared in the literature concern multiplier systems induced by Dirichlet characters (in the case of integral weights) or Dirichlet characters times the multiplier system of a power of $\eta(\tau)^{-2}\eta(2\tau)^{5}\eta(4\tau)^{-2}$ (in the case of half-integral weights). It is meaningful to deduce an explicit formula for $\dim_{\mathbb{C}}M_{k}(\Gamma_{0}(N),\chi_{1}\chi_{2})$ where $k\in\mathbb{Q}$ , $\chi_{1}$ is induced by an eta-quotient of fractional exponents and $\chi_{2}$ by a Dirichlet character.

Do generalized double coset operators exist for rational weights? Theorem 9.9 let us know for what numbers $l$ does the expression $T_{l}f$ make sense. The multiplier systems involved are required to be induced by eta-quotients of integral exponents. The question is: if the multiplier systems are induced by eta-quotients of fractional exponents, is there any nontrivial $T_{l}$ (i.e. $l>1$ )? All identities presented in Section 9.4 are based on Theorem 9.9. Thus if we can find any $T_{l}$ with $l>1$ in the case of fractional exponents, then, taking into account of Theorem 4.2, we would obtain identities involving $c_{f}(n)$ for infinitely many eta-quotients $f$ of fractional exponents. However, no such operator has been found up to now.

$L$ -functions and Euler products. According to Theorem 9.24, it is natural to associate an $L$ -function $L(f,s)=\sum_{l\in L_{f}}\frac{c_{l}}{l^{s}}$ with a weakly admissible eta-quotient $f$ . This association is different from the usual one $\sum_{n}\frac{c_{f}(n)}{n^{s}}$ . The identities presented in Section 9.4 can be rephrased as properties of this $L(f,s)$ . For instance, it has an Euler product. It is interesting to investigate $L(f,s)$ , e.g., the functional equations, the corresponding Weil’s theorem, etc.

Listing all weakly admissible eta-quotients. Open question: to find out all weakly admissible eta-quotients (if there exist only finitely many). Another open question: for each weakly admissible eta-quotient, to deduce identities like the one in Proposition 9.33 from (83).

Appendix A Usage of SageMath code

The SageMath code can be obtained from the repository [43]. It is an ipynb file which should be opened in a Sage Jupyter notebook. The code in the first two cells should be run after the file is opened. The first cell, which actually is the one in [44], contains Python/Sage functions dealing with eta-quotients among which we need those concerning the characters and Fourier coefficients of eta-quotients of fractional exponents. The second cell then contains Python/Sage functions on computing dimensions, on order-character relations, on checking identities and on generating tables in this paper.

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Dimension formulas for modular form spaces of rational weights, the classification of eta-quotient characters and an extension of Martin’s theorem

Abstract.

Key words and phrases:

2020 Mathematics Subject Classification:

1. Introduction

1.1. Dimension formulas

A special case of the Main Theorem.

1.2. Application I

Proposition 1.1.

1.3. Application II

Example.

Example.

Example.

1.4. Other results, structure of the paper and notations

2. Modular forms of rational weight

Definition 2.1.

Remark 2.2.

Remark 2.3.

Remark 2.4.

3. Divisors of modular forms

Definition 3.1.

Theorem 3.2.

Proof.

Remark 3.3.

Proposition 3.4.

Proof.

4. The main theorem: Dimension formulas for rational weights

Lemma 4.1.

Proof.

Theorem 4.2.

Proof.

Remark 4.3.

Remark 4.4.

5. Order-character relations

Proposition 5.1.

Proof.

Remark 5.2.

Example 5.3.

6. The classification of characters induced by eta-quotients

Lemma 6.1.

Proof.

Remark 6.2.

Remark 6.3.

Definition 6.4.

Lemma 6.5.

Proof.

Lemma 6.6.

Proof.

Definition 6.7.

Theorem 6.8.

Proof.

Corollary 6.9.

Proof.

Example 6.10.

Example 6.11.

Example 6.12.

Example 6.13.

Example 6.14.

7. Dimension formulas for weights 1/21/2, 11, 3/23/2 and small levels

Lemma 7.1.

Proof.

Remark 7.2.

7.1. The weight 1/21/2

Example 7.3.

Example 7.4.

Remark 7.5.

Remark 7.6.

7.2. The weight 11

7.3. The weight 3/23/2

8. Application: one-dimensional spaces and Eisenstein series of rational weights

Proposition 8.1.

Proof.

Corollary 8.2.

Definition 8.3 (Definition 3.1, [16]).

Theorem 8.4.

Proof.

Lemma 8.5.

Proof.

Proposition 8.6.

Proof.

7. Dimension formulas for weights $1/2$ , $1$ , $3/2$ and small levels

7.1. The weight $1/2$

7.2. The weight $1$

7.3. The weight $3/2$