Explicit Small Image Theorems for Residual Modular Representations

Abstract

Let $\rho$ f,$\lambda$ be the residual Galois representation attached to a newform f and a prime ideal $\lambda$ in the integer ring of its coefficient field. In this paper, we prove explicit bounds for the residue characteristic of the prime ideals $\lambda$ such that $\rho$ f,$\lambda$ is exceptional, that is reducible, of projective dihedral image, or of projective image isomorphic to A4, S4 or A5. We also develop explicit criteria to check the reducibility of $\rho$ f,$\lambda$ , leading to an algorithm that compute the exact set of such $\lambda$. We have implemented this algorithm in PARI/GP. Along the way, we construct lifts of Katz' $\theta$ operator in character zero, and we prove a new Sturm bound theorem.