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Questions about modular forms and related areas

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Why are Poincare series defined as they are?

We know the Poincare series are defined as the following: The $m^{th}$ Poincare series of weight $k$ for $\Gamma$ is: $$ P_{m}^{k} (z) = \sum_{(c,d)=1} (cz+d)^{-k} e^{2 \pi in(\tau z)}. $$ The defini …
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