Sign changes of a product of Dirichlet character and Fourier coefficients of half integral weight modular forms

Abstract

Let $f\in S_{k+1/2}(N,\chi )$ be a Hecke eigenform of half integral weight $k+1/2(k\ge 2)$ and the real nebentypus $\chi =\pm 1$ where the Fourier coefficients $a(n)$ are reals. We prove that the sequence $\{\chi (p^{\nu })a(t{p}^{2\nu }){\}}_{\nu \in \mathbb{N}}$ has infinitely many sign changes for almost all primes $p$ where $t$ is a squarefree integer such that $a(t)\ne 0$. The same result holds for the sequences of Fourier coefficients $\{a(t{p}^{2(2\nu +1)}){\}}_{\nu \in \mathbb{N}}$ and $\{a(t{p}^{4\nu }){\}}_{\nu \in \mathbb{N}}$.