I'm currently studying Hecke-operators and I'm curious how Hecke came up with them. The original definition he gave in his paper is $$\left( f \mid T_n\right) (z) = n^{k - 1} \sum_{ad = n, \, b \mod d, \, d > 0} d^{-k } f\left( \frac{az + b}{d}\right)$$ for a modular form $f$ of weight $k$. This is an averaging over the set of representatives of the action of $\mathrm{SL}_2(\mathbf{Z})$ on $M_n := \left\{ A \in \mathbf{Z}^{2 \times 2} : \det (A) = n \right\}$ (acting by left-multiplication), i.e. $$f \mid T_n = n^{\frac{k}{2} - 1} \sum_{\alpha \in \mathrm{SL}_2(\mathbf{Z}) \text{ \ } M_n} f \mid_k \alpha$$ where $\mid_k$ is the slash-operator. But why would someone consider such an 'averaging'? Is there anything one can hope for by averaging over $\mathrm{SL}_2(\mathbf{Z}) \text{ \ } M_n$?
More generally, why did Hecke consider operators of such kind?