Recovering Cusp Forms on GL(2) from Symmetric Cubes

Suppose π, π′ are cusp forms on GL(2), not of solvable polyhedral type, such that they have the same symmetric cubes. Then we show that either π, π′ are twist equivalent, or else a certain degree 36 L-function associated to the pair has a pole at s=1. If we further assume that the symmetric fifth power of π is automorphic, then in the latter case, π is icosahedral in a suitable sense, agreeing with the usual notion when there is an associated Galois representation.