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Let be σ Galois automorphism of a number field  E . We determine the cuspidal representations π of GL 2  (    ) such that σ(π) ⋍ π ⊗ ω for some Hecke character ω. 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.

On twists of cuspidal representations of GL(2)