Petersson inner product

In mathematics the Petersson inner product is an inner product defined on the space of entire modular forms. It was introduced by the German mathematician Hans Petersson.

Let ${\displaystyle \mathbb {M} _{k}}$ be the space of entire modular forms of weight ${\displaystyle k}$ and ${\displaystyle \mathbb {S} _{k}}$ the space of cusp forms.

The mapping ${\displaystyle \langle \cdot ,\cdot \rangle :\mathbb {M} _{k}\times \mathbb {S} _{k}\rightarrow \mathbb {C} }$,

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