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 $\mathfrak{p}$. 



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



Hilbert modular forms twist-equivalent to their conjugates
Use Galois Representations. By Schur's Lemma, the projective representation extends to G\_K. By Tate's theorem, this projective representation lifts to a genuine representation of G\_K. The restriction of this representation to G\_L is a twist of the original representation. Hence, after twisting, the HMF is invariant under Gal(L/K). Then use cyclic base change.


