On modular modℓ Galois representations with exceptional images

We give a parametrization of the possible Serre invariants (N,k,ν) of modular modℓ Galois representations of the exceptional types A4, S4, A5, in terms of local data attached to the fields cut out by the associated projective representations. We show how this result combined with certain global considerations leads to an effective procedure that will determine for a given eigenform f and prime ℓ whether a modℓ representation attached to f is exceptional. We illustrate with numerical examples.