We can define a function which has the modular transformation property of modular forms by averaging over the group action:
$$f(z)=\sum_{\gamma\in\langle T\rangle\backslash\Gamma}h(\gamma z)(cz+d)^{-k},$$
whenever this makes sense, where $T=\left(\begin{matrix}1&1\\0&1\end{matrix}\right)$ acts on $\Gamma$ by left-multiplication. If $h=1$ we recover the Eisenstein series. If $h(z)=\exp(2\pi i nz)$ with $n\in\Z$, which is indeed $1$-periodic, we produce what are called Poincaré series. A special case for which it is *not* necessary to divide by the left-action of $T$ is the function $f(z)=\sum_{\gamma\in\Gamma}(\gamma z)^{-n}(cz+d)^{-k}$, for suitable values of $n$.

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- Last edited by Andrew Sutherland on 2018-12-13 05:51:48

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