$\DeclareMathOperator\SL{SL}$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!