Take $F$ a number field, $\pi$ a cuspidal automorphic representation of $GL(3, \mathbb{A}\_F).$ Suppose $\pi \cong \pi\otimes \chi.$ Comparing central characters we see that $\chi$ must be cubic. 


Now suppose $\chi$ is a cubic character of $\mathbb{A}\_F^\times$. Is it known how to construct $\pi$ such that $\pi \cong \pi\otimes \chi$ (or whether it is possible to do so)?



Is there a known construction of Cuspidal representations of GL(3) isomorphic to their own twist?
Let $E / F$ be the cyclic cubic extension corresponding to $\chi$ by class field theory. Let $\sigma$ be a generator of $\operatorname{Gal}(E / F)$, and let $\psi$ be a character of $E^\times \backslash \mathbb{A}\_E^\times$ such that $\psi \ne \psi^\sigma$.


Then we can consider the automorphic induction of $\psi$ to an automorphic representation $\pi$ o $GL(3, \mathbb{A}\_F)$. This exists, by Example 1.1 of [Clozel's 1986 ICM survey](http://www.mathunion.org/ICM/ICM1986.1/Main/icm1986.1.0791.0797.ocr.pdf); it is cuspidal, because of our assumption on $\psi$ (Theorem 2.4(iv) of op.cit.); and it satisfies $\pi \cong \pi \otimes \chi$.


(If I understand Clozel's article correctly, these are actually the *only* examples of cuspidal $\pi$ such that $\pi \cong \pi \otimes \chi$.)


