On the Sizes of Gaps in the Fourier Expansion of Modular Forms

Abstract

AbstractLet be a cusp form with integer weight k ≥ 2 that is not a linear combination of forms with complex multiplication. For n ≥ 1, letConcerning bounded values of i f (n) we prove that for ∊ > 0 there exists M = M(∊, f ) such that Using results of Wu, we show that if f is a weight 2 cusp form for an elliptic curve without complex multiplication, then . Using a result of David and Pappalardi, we improve the exponent to for almost all newforms associated to elliptic curves without complex multiplication. Inspired by a classical paper of Selberg, we also investigate i f (n) on the average using well known bounds on the Riemann Zeta function.