The equidistribution of Fourier coefficients of half integral weight modular forms on the plane

Abstract

Let $f=\sum_{n=1}^{\infty}a(n)q^{n}\in S_{k+1/2}(N,\chi_{0})$ be a non-zero cuspidal Hecke eigenform of weight $k+\frac{1}{2}$ and the trivial nebentypus $\chi_{0}$ where the Fourier coefficients $a(n)$ are real. Bruinier and Kohnen conjectured that the signs of $a(n)$ are equidistributed. This conjecture was proved to be true by Inam, Wiese and Arias-de-Reyna for the subfamilies $\{a(t n^{2})\}_{n}$ where $t$ is a squarefree integer such that $a(t)\neq 0$. Let $q$ and $d$ be natural numbers such that $(d,q)=1$. In this work, we show that $\{a(t n^{2})\}_{n}$ is equidistributed over any arithmetic progression $n\equiv d\text{ mod }q$.