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

Number Theory;Representation Theory