Non-cuspidal Hecke eigenforms and Eisenstein series

It's a direct check that $$E_{2k}(z )=\frac{\zeta(1-2k)}{2}+\sum _{n=1}^{\infty }\sigma _{2k-1}(n)q^{n}}$$ is an eigenform for every Hecke operator $$T_n$$ with eigenvalue $$\sigma_{2k-1}(n)$$. Similarly, the Eisenstein series for $$Γ_0(N)$$ (at any cusp) are eigenfunctions for the Hecke operators $$T_n$$ with $$(n,N)=1$$.

These computation raises interests on non-cuspidal Hecke eigenforms. Do we know Eisenstein series are Hecke eigenforms more generally (something like Hilbert modular forms and p-adic modular forms) ? Is there a conceptual explanation?

More interestingly, how do we find all non-cuspidal Hecke eigenforms other than Eisenstein series?

• Define "the Eisenstein series for $\Gamma_0(N)$ at any cusp" ? To me an Ensenstein series $\in M_{2k}(\Gamma_0(N))$ is of the form $\sum_{\gamma \in S} \gamma'(z)^k$ where $S$ is a subset of $\langle T \rangle \setminus GL_2^+(\mathbb{R})$ where $T(z) = z+1$ such that $\langle T \rangle S \Gamma_0(N) =\langle T \rangle S$, or a linear combination of those. Not all of them are eigenforms. – reuns Nov 11 '18 at 19:40
• – sawdada Nov 11 '18 at 20:10
• Ok then my advice is similar to Paul Garrett's post : find the $S$ needed to relate my definition with the one in term of the congruences of $(n,m)$ (the effect of the Hecke operators on $S$ are easy to understand) – reuns Nov 11 '18 at 20:14
• Rankin has a series of papers "Diagonalizing Eisenstein series" where he studies these questions. If I recall correctly it is not true that the Eisenstein new subspace can be diagonalized wrt all the $T_n$, unlike the situation for cusp forms. – François Brunault Nov 11 '18 at 21:54