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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.

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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).

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  • $\begingroup$ What I know p.247 in Diamond's book : If $f$ is an eigenform $ \in S_k(\Gamma_0(N),\psi) , a_1=1$ and $\sigma$ is an embedding $\mathrm{Q}(\{a_n\}_{n \ge 1})\to \mathbb{C}$ then $f^\sigma = \sum_n \sigma(a_n) q^n$ is an eigenform in $S_k(\Gamma_0(N),\psi^\sigma)$. But how should we embed $f \mapsto f \otimes \chi$ into this ? $\endgroup$ – reuns Jun 11 '17 at 9:12
  • $\begingroup$ @reuns Actually one may define $\Gamma$ as the set of embeddings $\sigma : L \to \mathbf{C}$ such that $f^\sigma = f \otimes \chi_\sigma$ for some Dirichlet character $\chi_\sigma$. Then automatically $\Gamma$ is an abelian subgroup of $\mathrm{Aut}(L)$. See Ribet, Twists of modular forms and endomorphisms of abelian varieties. Note that $f \otimes \chi_\sigma$ is a newform with Nebentypus character $\psi \chi_\sigma^2$, hence we have in particular the equation $\psi^\sigma = \psi \chi_\sigma^2$. Does this answer your question? $\endgroup$ – François Brunault Jun 11 '17 at 15:47
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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).

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