Explicit Small Image Theorems for Residual Modular Representations

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