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][1]; 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 $\pi \cong \pi \otimes \chi$.) [1]: http://www.mathunion.org/ICM/ICM1986.1/Main/icm1986.1.0791.0797.ocr.pdf

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; 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 $\pi \cong \pi \otimes \chi$.)