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 $a(t)\ne 0$. 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 $n\equiv d\text{ mod }q$.

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