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 ∆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 Bk of n such that (Math Presents).