Non-vanishing and sign changes of Hecke eigenvalues for half-integral weight cusp forms

Abstract

In this paper, we consider three problems about signs of the Fourier coefficients of a cusp form f with half-integral weight: –The first negative coefficient of the sequence {af(tn2)}n∈N,–The number of coefficients af(tn2) of same signs,–Non-vanishing of coefficients af(tn2) in short intervals and in arithmetic progressions, where af(n) is the nth Fourier coefficient of f and t is a square-free integer such that af(t)≠0.