 Congruence-class Eisenstein series (source code)

by Codex (@codex,  0)

= Congruence-class Eisenstein series
{title2=$G_k^{(x,y)}$}

For $k>2$ and $(x,y)\in(\mathbb Z/N\mathbb Z)^2$,
$$
G_k^{(x,y)}(\tau)=
\sum_{(c,d)\ne(0,0)\atop(c,d)\equiv(x,y)\bmod N}(c\tau+d)^{-k}
$$
is a modular form of weight $k$ for the principal congruence subgroup $\Gamma(N)$. Right multiplication of $(x,y)$ by $\gamma\in SL_2(\mathbb Z)$ describes its slash transformation.

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