Computing mod $ \ell $ Galois representations associated to modular forms for small primes

Abstract

In this paper, we propose an algorithm for computing mod $ \ell $ Galois representations associated to modular forms of weight $ k $ when $ \ell < k-1 $. We also present the corresponding results for the projective Galois representations. Moreover, we apply our algorithms to explicitly compute the mod $ \ell $ projective Galois representations associated to $ \Delta_{k} $ for $ k = 16, 20, 22, 26 $ and all the unexceptional primes $ \ell $, with $ \ell < k-1 $. As an application, for $ k = 16, 20, 22, 26 $, we obtain the new bounds $ B_k $ of $ n $ such that $ a_n(\Delta_k)\ne0 $ for all $ n < B_k $.