Computing mod l Galois representations associated to modular forms for small primes

In this paper, we propose an algorithm for computing mod l Galois representations associated to modular forms of weight k when l < k-1. We also present the corresponding results for the projective Galois representations. Moreover, we apply our algorithms to explicitly compute the mod l projective Galois representations associated to [[DELTA].sub.k] for k = 16, 20, 22, 26 and all the unexceptional primes l, with l < k-1. As an application, for k = 16, 20, 22, 26, we obtain the new bounds [B.sub.k] of n such that an ([[DELTA].sub.k]) [not equal to] 0 for all n < [B.sub.k]. Keywords: modular forms; mod l Galois representations; unexceptional primes; polynomials associated to modular Galois representations; Fourier coefficients of modular forms Mathematics Subject Classification: 11F80, 11F33, 11G30, 11G18