Graded Algebras of Modular Forms and Sign Changes of Fourier Coefficients for Eta Quotients
Abstract
In this paper, for N=8,12,16,20,32, we determine the explicit structure of the graded algebra of modular forms for varGamma0(N) of both integral and half-integral weights, with all quadratic Dirichlet characters modulo N. We give explicit eta quotient generators and determine the relations among them. Using these structures, for a given weight, a level among those considered above, and a quadratic character, we obtain a basis for the corresponding space of modular forms consisting of eta quotients. Using these eta quotient bases, we obtain necessary and sufficient conditions for the periodicity of the signs of the Fourier coefficients of modular forms in three specific spaces. Finally, we determine all eta quotients in these spaces whose Fourier coefficient signs are periodic.
Related
- On sign patterns of Fourier coefficients of theta quotients
- Atkin--Lehner Quotients of Modular Curves at Primes of Bad Reduction
- Fourier Coefficients of Siegel-Eisenstein Series of Degree 2m and Weight m+1
- Total rings of quotients of higher dimensional Iwasawa algebras
- Fourier Coefficients of the Degenerate Eisenstein Series on Symplectic Groups
BibTeX
@misc{gan2026graded,
title = {{Graded Algebras of Modular Forms and Sign Changes of Fourier Coefficients for Eta Quotients}},
author = {Jianwen Gan},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.18339},
eprint = {2609.18339},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18339},
}
RIS
TY - UNPB
TI - Graded Algebras of Modular Forms and Sign Changes of Fourier Coefficients for Eta Quotients
AU - Jianwen Gan
PY - 2026
DA - 2026/10/25
PB - arXiv
DO - 10.48550/arxiv.2609.18339
UR - https://arxiv.org/abs/2609.18339
AB - In this paper, for N=8,12,16,20,32, we determine the explicit structure of the graded algebra of modular forms for varGamma_0(N) of both integral and half-integral weights, with all quadratic Dirichlet characters modulo N. We give explicit eta quotient generators and determine the relations among them. Using these structures, for a given weight, a level among those considered above, and a quadratic character, we obtain a basis for the corresponding space of modular forms consisting of eta quotients. Using these eta quotient bases, we obtain necessary and sufficient conditions for the periodicity of the signs of the Fourier coefficients of modular forms in three specific spaces. Finally, we determine all eta quotients in these spaces whose Fourier coefficient signs are periodic.
ER -