Abstract. Given an odd, semisimple, reducible, 2-dimensional mod l Galois representation, we investigate the possible levels of the modular forms giving rise to it. When the representation is the direct sum of the trivial character and a power of the mod l cyclotomic character, we are able to characterize the primes that can arise as levels of the associated newforms. As an application, we determine a new explicit lower bound for the highest degree among the fields of coefficients of newforms of trivial Nebentypus and prime level. The bound is valid in a subset of the primes with natural (lower) density at least 3/4.

Let k and N be positive integers with k ≥ 2 even. In this paper we give general explicit upper-bounds in terms of k and N from which all the residual representations ρ f,λ attached to non-CM newforms of weight k and level Γ 0 (N ) with λ of residue characteristic greater than these bounds are "as large as possible". The results split into different cases according to the possible types for the residual images and each of them is illustrated on some numerical examples.