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I am beginning to study modular forms and I came across an inequality defining the bound for the degree of extension of the field of coefficients of a modular form $f\in S_k(\Gamma_1(N))$. This goes as follows

$$[K_f/\mathbb{Q}]\leq \text{rank}_{\mathbb{Z}}\mathbb{T}(S_k^{\text{}new}(\Gamma_1(N)))$$ where $\mathbb{T}$ is the algebra of Hecke operators acting on $S_k^{\text{}new}(\Gamma_1(N))$.

Is this true for any level and weight? If it is can anyone give or refer me to the proof of this?

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  • $\begingroup$ Where did you come across this inequality? And what do you mean by $\mathbb{T}$? $\endgroup$
    – davidlowryduda
    Commented Jun 10 at 16:01
  • $\begingroup$ Sorry for my unclear question earlier. I have edited it now. By \mathcal{T}, I meant the algebra of Hecke operators. You can find it here math.leidenuniv.nl/~pbruin/coefficients.pdf. It doesn't gives a proof or mentions any reference though. $\endgroup$ Commented Jun 10 at 21:06
  • $\begingroup$ I'm sure there's a better way, but this implicitly follows from methods of computing modular forms using modular symbols. See for instance Algorithm 9.14 at wstein.org/books/modform/modform/newforms.html. $\endgroup$
    – davidlowryduda
    Commented Jun 13 at 19:56

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Yes, this is true regardless of level and weight, provided that $f$ is a normalized newform. The point is that $T_n \in \mathbb{T}\otimes \mathbb{Q} \longmapsto a_n(f) \in K_f$ is a surjective ring homomorphism.

It is (much) more difficult to show that $\mathbb{T}\otimes \mathbb{Q}$ is a finite dimensional $\mathbb{Q}$-algebra (in fact, that $\mathbb{T}$ is a finite free $\mathbb{Z}$-algebra), and that newforms correspond exactly to ring homomorphisms $\mathbb{T} \rightarrow \mathbb{C}$ (whence, for instance, the fact that the Galois-conjugate of a newform is a newform).

One way to go about this is to use modular symbols as hinted in davidlowryduda’s comments: you construct a finitely generated abelian group of modular symbols $S := H^1_{par}(\Gamma_1(N),V_{k-2}(\mathbb{Z}))$ and an isomorphism $S \otimes_{\mathbb{Z}} \mathbb{C} \simeq \mathcal{S}_k(\Gamma_1(N))$. Then you can show that under this isomorphism, the action of $\mathbb{T}$ on the RHS comes from an action of $\mathbb{T}$ on $S$.

I suggest Gabor Wiese’s notes on this topic https://math.uni.lu/~wiese/notes/MFII.pdf.

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