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.

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