Fourier coefficients of automorphic L-functions

Abstract

We prove two unconditional results on Fourier coefficients of modular forms: Let [Formula: see text] be a Hecke eigen cusp form of weight [Formula: see text] and level [Formula: see text] ([Formula: see text] square free), and [Formula: see text] be the normalized Hecke eigenvalues. Let [Formula: see text] be the least prime [Formula: see text] such that [Formula: see text]. Then we show that in a family of modular forms of weight [Formula: see text] and level [Formula: see text], except for a density zero set, [Formula: see text] for some [Formula: see text]. Second, we show that outside a density zero set, a modular form is determined by Fourier coefficients up to [Formula: see text] for some [Formula: see text].