Weight $k$ slash-action (reviewed)

Let $f$ be a function from $\mathcal{H}$ to $\mathbb{C}$. We define the weight $k$ slash-action $f|_k\gamma$ to be the action of $\gamma=\left(\begin{matrix}a&b\\c&d\end{matrix}\right)\in\textrm{SL}_2(\mathbb{R})$ given by \[ f|_k\gamma(z)=\det(\gamma)^{k/2}(cz+d)^{-k}f(\gamma z). \]

Note that, if $f$ is a modular form with multiplier system $v$, the modularity condition is equivalent to \[ f|_k\gamma(z)=v(\gamma)f(z). \]