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arXiv · 2606.04203

Enumeration of modular forms for $\Gamma_1(N)$

Timothy Huber·Jeffery Opoku·Dongxi Ye
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Abstract

This paper considers holomorphic modular forms for $\Gamma_1(N)$ of integral weight of the form $$f^{(N)}_{\mathbf a}(\tau) =q^{s} (q^{N};q^{N})_{\infty}^{a_0}\prod_{j=1}^{\lfloor N/2 \rfloor}(q^j,q^{N-j};q^N)_\infty^{a_j}, \quad \mathbf a = (a_1, \ldots, a_{\lfloor N/2 \rfloor}),$$ for fixed $a_0=2k \in 2 \Bbb Z_{\ge 0}$. We show that the number of relevant exponent vectors $\mathbf a$ is finite and characterize them in terms of the $\mathbb{Q}$-rational cuspidal divisor class group of $X_{1}(N)$. Effective procedures are given for counting the admissible exponents by enumerating the corresponding polytopes. This leads to formulas for the number of exponent vectors in terms of quasipolynomials in $k$.

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Timothy Huber, Jeffery Opoku, Dongxi Ye. 2026-06-02. Enumeration of modular forms for $\Gamma_1(N)$. https://arxiv.org/abs/2606.04203

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