Sign changes of a product of Dirichlet character and Fourier coefficients of half integral weight modular forms
Let $f\in S_{k+1/2}(N,\chi) be a Hecke eigenform of half integral weight $k+1/2\(k\geq 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(tp^{2\nu}}_{\nu\in\N} has infinitely many sign changes for almost all primes $p$ where $t$ is a squarefree integer such that $a(t)\neq 0$ The same result holds for the sequences of Fourier coefficients {a(tp^{2(2\nu+1)}_{\nu\in\N} and {a(tp^{4\nu}}_{\nu\in\N}