It's a direct check that ${\displaystyle E_{2k}(z )=\frac{\zeta(1-2k)}{2}+\sum _{n=1}^{\infty }\sigma _{2k-1}(n)q^{n}}$ 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?