HECKE-SIEGEL TYPE THRESHOLD FOR SQUARE-FREE FOURIER COEFFICIENTS : AN IMPROVEMENT (Analytic and Arithmetic Theory of Automorphic Forms)

Abstract

We prove that if f is a non zero cusp form of weight k on Gamma_{0}(N) with character chi such that N/(conductor chi) is square-free, and epsilon>0 is given, then there exists a square-free nleq k^{3+epsilon}N^{7/2+epsilon} such that a(f, n)neq 0. This significantly improves the already known existential and quantitative result from previous works.