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).

The Galois group of a non-Galois field extension is usually defined to be the Galois group of its Galois closure. This definition coincides with the Galois group of a polynomial, if we take the irreducible polynomial of a generator of the field. It also is the direct analogue, via the Galois group / fundamental group dictionary, of the monodromy group of a covering space, sheaf, or vector bundle with flat connection.

I think it is expected that the coefficient fields of modular forms are "usually" not Galois. For instance it is conjectured (by Maeda) that cusp forms of level 1 have an S_n Galois group, n the dimension of the space of cusp forms, and hence are not Galois for n > 2 (because S_n does not act simply transitively on n objects for n > 2).