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Let $N \geq 11$ be an integer and consider the basis of Eisenstein series for $M_2(\Gamma_0(N))$ described in Theorem $4.6.2$ of Diamond--Shurman's book. Pick and Eisenstein series $F$ in this basis. One can view $F$ as a rule on pairs $(E,\phi)$ where $E$ is an elliptic curve (over some $\mathbb{Z}[\frac{1}{N}]$-algebra) and $\phi$ is an isogeny of degree $N$.

My question : what is this rule explicitly in terms of the data $\phi$, $\psi$ and $t$ defining $F$ as in Diamond--Shurman ? For example we can take $N$ prime and $F = \frac{N-1}{24} + \sum_{n \geq 1} \sigma'_1(n)q^n$ where $\sigma'_1(n) = \sum_{d \mid n, gcd(d,N)=1} d$ (the unique Eisenstein series of $M_2(\Gamma_0(N))$).

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