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In this paper we prove the following conditional result: Let F be a numberfield, and pi a cusp form on GL(2)/F which is not solvable polyhedral. Assumethat all the symmetric powers sym^m(pi) are modular, i.e., define automorphicforms on GL(m+1)/F. If sym^6(pi) is cuspidal, then all the symmetric powers arecuspidal, for all m. Moreover, sym^6(pi) is Eisenteinian iff sym^5(pi) is anabelian twist of the functorial product of pi with the symmetric square of acusp form pi' on GL(2)/F.

Remarks on the symmetric powers of cusp forms on GL(2)