# Looking for clarification on a reference regarding Eisenstein series and eigenforms.

William Stein has a lovely book Modular Forms, A Computational Approach, and I'd like some clarification on one of the theorems from the section on Eisenstein series. Theorem 5.8 states that for characters $$\chi$$ and $$\psi$$ with conductors $$L$$ and $$R$$, respectively, that the Eisenstein series $$E_{k,\chi,\psi}(z)$$ is a modular form in $$M_k(RL,\chi\psi)$$ (I've elided a couple extra conditions for clarity).

Now, Theorem 5.10 says that $$E_{k,\chi,\psi}(z) \in M_k(RL)$$ are eigenforms for all Hecke operators. Is the intention of this that the form is only an eigenform when $$\chi\psi=1$$, or am I misunderstanding the notation? As far as I can tell, this is the only place in the entire book where this particular modular forms space notation $$M_k(N)$$ is used, so I don't know exactly what it refers to.

Thanks!

• I think generally the notation is $M_k(N) := \bigoplus_{\chi \in (\mathbb{Z}/N \mathbb{Z})^* }M_k(N, \chi)$, so no you don't require $\chi \psi=1$. – ramanujan_dirac Jun 6 at 14:44
• The Hecke operators are not the same for $M_k(\Gamma_0(N))$ and $M_k(\Gamma_0(N),\chi)$, to unify them you need to go in $M_k(\Gamma_1(N))$, each of them have their eigenforms – reuns Jun 6 at 16:20
• Thanks @ramanujan_dirac and reuns. I suspected this was the case, but didn't want to rely on it without knowing for sure. – loofthri Jun 9 at 12:17