Let $P$ be a prime number, and let $k >2$ be a positive even integer. Now I want to compute the Fourier expansion of the Eisenstein series of $\Gamma_0(p)$ at the cusp $0$, i.e., $$ E_{k, 0}(z) = \sum_{\gamma \in \Gamma_0 \backslash \Gamma} j( S \gamma S^{-1}, z)^k. $$ Here $\Gamma=\Gamma_0(P)$, $\Gamma_0$ is the stabilizer of the cusp $0$ in $\Gamma$, and $S=\begin{pmatrix}0&{-1}\\1&0\\ \end{pmatrix}$.
I only see one example in the text to compute fourier expansion, which is very different with this one. Basically I have no idea about how to handle this. So where can I find similar examples of calculation?