Sign of Fourier coefficients of half-integral weight modular forms in arithmetic progressions

TL;DR

This paper investigates the signs of Fourier coefficients of half-integral weight modular forms in arithmetic progressions, providing lower bounds for the count of coefficients exceeding certain thresholds, regardless of eigenform status.

Contribution

It introduces new bounds on the distribution of Fourier coefficient signs in arithmetic progressions for a broad class of half-integral weight modular forms.

Findings

Established lower bounds for the number of coefficients with sign positivity.

Extended results to non-eigenform cusp forms.

Provided quantitative estimates for coefficients exceeding a power threshold.

Abstract

Let $f$ be a half-integral weight cusp form of level $4N$ for odd and squarefree $N$ and let $a(n)$ denote its $n^{\rm th}$ normalized Fourier coefficient. Assuming that all the coefficients $a(n)$ are real, we study the sign of $a(n)$ when $n$ runs through an arithmetic progression. As a consequence, we establish a lower bound for the number of integers $n\le x$ such that $a(n)>n^{-\alpha}$ where $x$ and $\alpha$ are positive and $f$ is not necessarily a Hecke eigenform.