Graded Algebras of Modular Forms and Sign Changes of Fourier Coefficients for Eta Quotients
Year: 2026
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Author: Gan, Jianwen
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213512141