On twists of cuspidal representations of GL(2)

Let sigma be a Galois automorphism of a number held E. We determine the cuspidal representations pi of GL(2)(A(E)) such that sigma (pi) similar or equal to pi circle times omega for some Hecke character omega. Our result is applied to extend the descent criterion of the base change theorem for GL(2) from cyclic extensions of prime degree to arbitrary cyclic extensions.