  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.
