The Equidistribution of Fourier Coefficients of Half Integral Weight Modular Forms on the Plane

Let Letf=∑n=1∞a(n)qn∈Sk+1/2(N,χ0)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\rm{Let}}\;f = \sum\limits_{n = 1}^\infty {a(n){q^n} \in {S_{k + 1/2}}(N,{\chi _0})} $$\end{document} be a nonzero cuspidal Hecke eigenform of weight k+12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k + {1 \over 2}$$\end{document} and the trivial nebentypus χ0, 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(tn2)}n, where t is a squarefree integer such that a(t) ≠ 0. Let q and d be natural numbers such that (d, q) = 1. In this work, we show that {a(tn2)}n is equidistributed over any arithmetic progression n ≡ d mod q.