Transformation property for modular forms (reviewed)

A modular form $f$ satisfies a transformation property for a group $\Gamma$, which relates the value of $f$ at a point in its domain with the value of $f$ at the point transformed under the action of $\Gamma$ on the domain.

For example in the classical case of $\Gamma = \SL(2,\Z)$, a modular form $f:$$\mathcal{H}$$\rightarrow\C$ satisfies the transformation \[ f(\gamma z) = f\left(\frac{az+b}{cz+d}\right)= v(\gamma)(cz+d)^kf(z), \]
for all $z\in\mathcal{H}$, $\gamma=\left(\begin{array}{ll}a&b\\c&d\end{array}\right)\in$$\SL_2(\Z)$ and multiplier system $v$.