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) 6= 0, we show that for all but finitely many primes p the sequence (a(tp))m has infinitely many signs changes. Moreover, we prove similar (partly conditional) results for arbitrary cusp forms f which are not necessarily Hecke eigenforms.