TITLE: Hilbert modular forms twist-equivalent to their conjugates
QUESTION [5 upvotes]: Let $L / K$ be a solvable (or cyclic) Galois extension of totally real fields, and let $f$ be a Hilbert modular newform over $L$.
Suppose that, for every $\sigma \in Gal(L / K)$, the conjugate newform $f_\sigma$ is twist-equivalent to $f$, i.e. there exists a Hecke character $\chi_\sigma$ of $L$ such that $a_{\sigma(\mathfrak{p})}(f) = \chi_{\sigma}(\mathfrak{p}) a_\mathfrak{p}(f)$ for all but finitely many primes $\mathfrak{p}$ of $L$, where $a_{\mathfrak{p}}(f)$ is the Hecke eigenvalue at $\mathfrak{p}$. 

Does it follow that $f$ is twist-equivalent to the base-change to $L$ of a Hilbert modular form over $K$? 

I'm happy to assume that $f$ is non-CM, and that all weights of $f$ are $\ge 2$, if that helps.
(Note that Galois is acting on $L$ here, not on the coefficients of $f$ -- this is not the same setup as the Ribet--Momose theory of "inner twists".)

REPLY [2 votes]: I'm going to add an answer to my own question, not because there's anything wrong with user94346's answer, but because I found a paper in the literature treating exactly this question:
Erez Lapid and Jonathan Rogawski, On twists of cuspidal representations of GL(2). Forum Math. 10 (1998), no. 2, 175–197. DOI: 10.1515/form.10.2.175