In Shimura's 1978 paper he introduces Hilbert Eisenstein Series $E_k(\eta , \psi ) \in M_k(\mathfrak {ab}, \eta \psi)$ such that $$ c(\mathfrak m, E_k(\eta , \psi ))= \sum _{\mathfrak r | \mathfrak m } \eta \bigg(\frac{\mathfrak m}{\mathfrak r}\bigg)\psi (\mathfrak r ) N\mathfrak r ^{k-1}$$ for all non-zero ideals $\mathfrak m $ of $\mathcal O_F$.
I want a neat description of how Hecke operators $T_\mathfrak p$ and $U_\mathfrak p $ act on this Eisenstein series. Any reference would be appreciated. Thank you.