$\DeclareMathOperator\SL{SL}$It is well-known that the cuspidal (or discrete) part of $L^2(\SL(2,\mathbb{Z})\backslash{\SL(2,\mathbb{R})})$ decomposes into irreducible representations of $\SL(2,\mathbb{R})$. One can see Theorem 2.6 of Gelbart's book Automorphic Forms on Adele Groups.

$L^2_{\text{cusp}}(\SL(2,\mathbb{Z})\backslash{\SL(2,\mathbb R)})=\bigoplus_{\pi\in\widehat{SL(2,\mathbb{R})}}m_{\pi}\cdot\pi$.

My question is which $m_\pi$ is nonzero and what is the formula?

In the Gelbart's book (Theorem 2.10), for the discrete series $\pi_k$, $m_{\pi_k}=\dim S_k(\SL(2,\mathbb{Z}))$, the dimension of cusp forms of weight $k$. How about the other $m_{\pi}$'s? The principal series may also be related to the dimension of wave forms.

Is there any further (complete) result for the decomposition or for a general pair of $\Gamma\subset G$, a lattice in a real Lie group? The corresponding results for the adèle groups and automorphic representations are also welcome!