Distribution of integral Fourier Coefficients of a Modular Form of Half Integral Weight Modulo Primes

Abstract

Recently, Bruinier and Ono classified cusp forms $f(z):={\sum }_{n=0}^{\infty }{a}_f(n)q^n\in S_{\lambda +1/2}({\Gamma }_0(N),\chi )\cap \mathbb{Z}[[q]]$ that does not satisfy a certain distribution property for modulo odd primes $p$. In this paper, using Rankin-Cohen Bracket, we extend this result to modular forms of half integral weight for primes $p\ge 5$. As applications of our main theorem we derive distribution properties, for modulo primes $p\ge 5$, of traces of singular moduli and Hurwitz class number. We also study an analogue of Newman's conjecture for overpartitions.