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.



When is a Fourier coefficient field Galois?
As Will Sawin points out, it is not true in general that the Hecke field of a modular form is a Galois number field. The first example in weight $2$ and level $\Gamma\_0(N)$ appears at $N=41$: the space $S\_2(\Gamma\_0(41))$ has dimension $3$ and is generated by the conjugates of a newform $f$ with Hecke field $L=\mathbf{Q}(\alpha)$ with $\alpha$ being a root of the irreducible polynomial $X^3+X^2-5X-1$. The number field $L$ is not Galois, and its Galois closure has Galois group $\mathfrak{S}\_3$.


In general, let $f \in S\_k(\Gamma\_1(N),\varepsilon)$ be a newform without CM, and let $L$ be the Hecke field of $f$. Let $\Gamma$ be the group of inner-twists of $f$, consisting of those automorphisms $\sigma$ of $L$ satisfying $f^\sigma = f \otimes \chi\_\sigma$ for some Dirichlet character $\chi\_\sigma$. It is known that $L/L^\Gamma$ is a finite abelian extension and that $L^\Gamma=\mathbf{Q}(\{a\_p^2/\varepsilon(p)\}\_{p \nmid N})$ is a totally real number field. But $L^\Gamma$ need not be Galois over $\mathbf{Q}$, as the preceding example already shows ($\mathrm{Aut}(L)$ is clearly trivial in this case).


