Maass cusp forms

Question

I want to show that Space of Maass cusp forms is Hilbert space with Petersson inner product. Where Maass cusp form is defined to be smooth functions on upper half plane which are

1. Eigen vectors of k-weighted hyperbolic Laplace operator.
2. Satisfies transformation law : $f(Mz)=e^{i arg(cz+d)k}f(z)$ for all $z \in \mathbb{H}$ and M in $SL_2(Z)$. Here c,d are bottom row entries of M.
3. As y tends to infinity f decays exponentially.

My idea: I want to show that its Hilbert space with petersson inner product. So its enough to show that it is closed subspace of Hilbert space $L^2(F)$, where F is fundamental domain of $SL_2(Z)$. Any hint/ help would be appreciated.