$\DeclareMathOperator\SL{SL}$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 $\SL(2,\mathbb{R})$. As far as we know, the multiplicities of discrete series or principle series are just the dimensions of holomorphic cusp forms or Maass cusp forms (see Theorem 2.10 of Gelbart's book Automorphic Forms on Adele Groups).

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!