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I would like to understand how the Hecke operator for the SL(2,Z) group works. In particular its action on, for example a constant function. Would the following developpement be right?

$T_n(f(\tau)) := n^{k-1}\sum_{ad=n, d>0} \frac{1}{d^k}\sum_{b=0}^{d-1} f(\frac{a\tau +b}{d})$

Thus,

$T_n(1) = \frac{1}{n}\sum_{ad=n, d>0} \sum_{b=0}^{d-1} 1$

$T_n(1) = \frac{1}{n}\sum_{ad=n, d>0}d-1$

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