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!