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)?