Level of Hecke eigenforms

algebraic-number-theorymodular-formssolution-verification

I have been reading some notes and books on modular forms and frequently meet the sententence "Let $f$ be a weight 2, level $N$ Hecke eigenform". But some of them have not make it clear whether $f \in S_2(\Gamma_0(N))$ or $f \in S_2(\Gamma_1(N))$. It makes me feel quite confused.

So my question is: is there any usual convention on that?

My attempts:

  1. Since we often define the Hecke operators $\langle d \rangle$ and $T_n$ (where $n$ is an integer and $d \in (\mathbb{Z}/N\mathbb{Z})^{\times}$) as linear operators on $S_2(\Gamma_1(N))$ (as in the book by Fred Diamond and Jerry Shurman) and Hecke eigenforms are defined as the "eigenvectors" of all operators in the sub-$\mathbb{C}$-algebra in $\mathrm{End}_{\mathbb{C}}(S_2(\Gamma_1(N)))$ generated by operators $\{\langle d \rangle, T_p \}_{d \in (\mathbb{Z}/N\mathbb{Z})^{\times}, p \, \text{prime}}$, the Hecke eigenforms are indeed cuspidal forms in $S_2(\Gamma_1(N))$.

  2. However, we see that in particular, a Hecke eigenform lies in all eigenspaces of diamond operators
    $$
    S_2(N, \chi) := M_2(N, \chi) \cap S_2(\Gamma_1(N)) := \{ f \in M_2(\Gamma_1(N)) : \langle d \rangle f = \chi(d) f, \forall d \in (\mathbb{Z}/N\mathbb{Z})^{\times} \} \cap S_2(\Gamma_1(N)) ,
    $$
    where $\chi: (\mathbb{Z}/N\mathbb{Z})^{\times} \rightarrow \mathbb{C}$ is a Dirichlet character. In Exercise 4.4.3(a) of Fred Diamond and Jerry Shurman, we have shown that
    $$
    M_2(N, \mathbf{1}_N) = M_2(\Gamma_0(N)).
    $$
    Hence as a Hecke eigenform, $f$ ought to be an element in $S_2(N, \mathbf{1}_N) := M_2(N, \mathbf{1}_N) \cap S_2(\Gamma_1(N)) = S_2(\Gamma_0(N))$. Hence actually as a Hecke eigenform, $f$ also has level $\Gamma_0(N)$. So we do not need to distinguish the level "$\Gamma_0$" or "$\Gamma_1$".

So are my attempts correct?

All my knowledges on modular forms are self-teached, so I'm sorry if the question is too trivial or I had made some silly mistakes.

Thank you all for answering and commenting!

Best Answer

Any $S_k(\Gamma_1(N))$ eigenform is in $S_k(\Gamma_0(N),\chi)$ for some $\chi\bmod N$.

This is because the projection map $\sum_{d\bmod N} \overline{\chi(d) }\langle d\rangle: S_k(\Gamma_1(N))\to S_k(\Gamma_0(N),\chi)$ commutes with the Hecke operators.

Whence some authors mean $f\in S_k(\Gamma_0(N))$, some others mean $f\in S_k(\Gamma_0(N),\chi)$, which doesn't make a big difference in most cases.

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