Recovering Cusp forms on GL(2) from Symmetric Cubes

Suppose $\pi$, $\pi'$ are cusp forms on GL$(2)$, not of solvable polyhedral type, such that they have the same symmetric cubes. Then we show that either $\pi$, $\pi'$ 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 $\pi$ is automorphic, then in the latter case, $\pi$ is icosahedral in a suitable sense, agreeing with the usual notion when there is an associated Galois representation.