Sign of Fourier coefficients of half-integral weight modular forms in arithmetic progressions

Abstract

Let f be a half-integral weight cusp form of level 4N for odd and squarefree N and let a(n) denote its nth normalized Fourier coefficient. Assuming that all the coefficients a(n) are real, we study the sign of a(n) when n runs through an arithmetic progression. As a consequence, we establish a lower bound for the number of integers $$n\leqslant x$$ such that $$a(n)>n^{-\alpha }$$ where x and $$\alpha $$ are positive and f is not necessarily a Hecke eigenform.