Fourier coefficients of automorphic L-functions

We prove two unconditional results on Fourier coefficients of modular forms: Let f be a Hecke eigen cusp form of weight k and level N (N square free), and ?(f)(n) be the normalized Hecke eigenvalues. Let n(f) be the least prime p such that ?(f)(p) < 0. Then we show that in a family of modular forms of weight k and level N, except for a density zero set, n(f) << (log k(2)N)(a) for some a > 0. Second, we show that outside a density zero set, a modular form is determined by Fourier coefficients up to O((log k(2)N)(a)) for some a > 0.