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

Abstract

Let $f$ be a half-integral weight cusp form of level $4N$ for odd and squarefree $N$ and let $a(n)$ denote its $n^{\mathrm{th}}$ 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.