# Level of Hecke eigenforms

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!

• It's also quite possible they mean $\Gamma(N)$. Or any other $\Gamma \supset\Gamma(N)$. I don't think there is a strong convention either way (although $\Gamma_1(N)$ is most likely). It's pretty common to hear someone define a modular form of level $N$ only for someone in the audience to ask them if they mean with character ($\Gamma_1$) or not ($\Gamma_0$). Commented Apr 23, 2021 at 14:26
• Also, you really don't need to apologise about asking questions! For the record, none of your questions so far on this topic have been silly or trivial. Commented Apr 23, 2021 at 14:30
• @Mathmo123 Thank you for your comments and especially for your encouragement! :) Commented Apr 23, 2021 at 15:15

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.
• Thank you for your answer! So it seems that the space with the nebentypus $\chi$, i.e. $S_k(\Gamma_0(N), \chi) := \{ f \in S_k(\Gamma_0(N)) : \langle d \rangle f = \chi(d) f, \forall d \in (\mathbb{Z}/N\mathbb{Z})^{\times} \}$ is a subspace of $S_k(\Gamma_0(N))$ and hence I can regard the character $\chi$ as some additional information for the eigenform $f$? Is my understanding correct? Besides, I'm actually reading things on modularity lifting, and hoping that the differences on $f \in S_k(\Gamma_0(N))$ and $f \in S_k(\Gamma_0(N), \chi)$ will not make a big difference on their Galois reps. Commented Apr 23, 2021 at 12:51
• If $\chi$ is not real then $f \in S_k(\Gamma_0(N), \chi)$ doesn't have rational coefficients. I'm a bit unsure for the quadratic character. Commented Apr 23, 2021 at 12:53
• Thank you! Yet sorry for that I'm not quite familiar with the meaning of quadratic twist of the underlying elliptic curve. And I'm a bit confused that if $S_k(\Gamma_0(N), \chi)$ is indeed a subspace of $S_k(\Gamma_0(N))$, where are the real differences between the two conventions (as you mentioned in the post). It seems that by writing $f \in S_k(\Gamma_0(N), \chi)$, we only want to adress the further information on $f$, namely the nebentypus $\chi$ of $f$? Commented Apr 23, 2021 at 13:03
• $$S_k(\Gamma_1(N))=\bigoplus_\chi S_k(\Gamma_0(N), \chi)$$ this follows from the projection map $\frac1{\varphi(N)}\sum_{d\bmod N} \overline{\chi(d) }\langle d\rangle$. For $f\in S_k(\Gamma_0(N),\phi)$ we have $\sum_n a_n(f)\chi(n)e^{2i\pi n z} \in S_k(\Gamma_0(N),\phi \chi^2)$ (or $\phi \overline{\chi}^2$ I should redo the calculation) so the $S_k(\Gamma_0(N),\phi \chi^2)$ are tightly related for different $\chi$, though the Galois representation will be a bit different. Commented Apr 23, 2021 at 13:10
• My quadratic twist comment was a mistake, if $\sum_{n\ge 1} a_n(f) e^{2i\pi nz}\in S_2(\Gamma_0(N))$ is an eigenform with rational coefficients then its L-function is that of an elliptic curve $y^2=x^3+ax+b$ and $\sum_{n\ge 1} (\frac{n}{N}) a_n(f) e^{2i\pi nz}\in S_2(\Gamma_0(N))$ is that of $\pm N y^2=x^3+ax+b$. The map $y^2=x^3+ax+b\to N y^2=x^3+ax+b$ has an easy interpretation in term of the Galois representation of the elliptic curve and its L-function. Commented Apr 23, 2021 at 13:13