On the sizes of gaps in the Fourier expansion of modular forms

Let f = E ' 1 af (n)q' be a cusp form with integer weight k >= 2 that is not a linear ncombination of forms with complex multiplication. For n >= 1, let i (f)(n) = max{i : a(f)(n + j) = 0 for all 0 <= j <= i} if a(f)(n) = 0, otherwise. Concerning bounded values of i (f)(n) we prove that for epsilon > 0 there exists M = M(epsilon, f) such that #{n <= x : i(f) (n) <= M} >= (1-epsilon)x. Using results of Wu, we show that if f is a weight 2 cusp form for an elliptic curve without complex multiplication, then i (f)(n) << (f,c) n5314. Using a result of David and Pappalardi, we improve the exponent to (1)/(3) for almost all newforms associated to elliptic curves 3 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.