$\DeclareMathOperator\SL{SL}$It is well-known that the cupsidal (or discrete) part $L^2(\SL(2,\mathbb{Z})\backslash{\SL(2,\mathbb{R})})$ decomposes into irreducible representations of $\SL(2,\mathbb{R})$. As far as we know, the multiplicities of discrete series or complementary series are just the dimensions of cusp forms or wave forms (see Theorem 2.10 of S. Gelbart's book in 1975).

Is there any further (complete) result for the decomposition or for any specific/general case? The corresponding results for the adèle groups and automorphic representations are also welcome!