Let $f$ be a modular form with $q$-coefficients $a_n$, and let $L=\mathbf Q(a_n:n\ge 0)$ be the Fourier coefficient field.

Does anyone know of any necessary or sufficient conditions for $L/\mathbf Q$ to be a Galois extension?

I know that by Momose, if $f$ is a non-CM newform with weight $\ge 2$ then you can embed the group $\Gamma$ of inner-twists into $\operatorname{Aut}(L/\mathbf Q)$ and then $L/L^\Gamma$ is a Galois extension, and I have seen one paper refer to "$\operatorname{Gal}(L^\Gamma/\mathbf Q)$", which would suggest that $L/\mathbf Q$ is Galois, but since they don't justify referring to the extension as Galois I've decided that either they've made a mistake, or they know something I don't.