Sign changes of coefficients of half integral weight modular forms

Abstract

For a half integral weight modular form $f$ we study the signs of the Fourier coefficients $a(n)$. If $f$ is a Hecke eigenform of level $N$ with real Nebentypus character, and $t$ is a fixed square-free positive integer with $a(t)\ne 0$, we show that for all but finitely many primes $p$ the sequence $(a(t{p}^{2m}){)}_m$ has infinitely many signs changes. Moreover, we prove similar (partly conditional) results for arbitrary cusp forms $f$ which are not necessarily Hecke eigenforms.