Projective Linear Groups as Galois Groups overQvia Modular Representations

We give an algorithm for the determination of the finitely many primes such that the image of the modular Galois representations attached to a weight 2 newform on Γ0(N) without complex multiplication or inner twists may not be “as large as possible". We apply the algorithm to suitable newforms and we obtain the realization of the groups PSL(2,Fℓ2),PGL (2,Fℓ3) andPSL (2,Fℓ4) as Galois groups overQfor high density sets of primes.