Abstract.The trace formula for SL{2,Z) can be developed for vector-valued functions which satisfy an automorphic condition involving a group representation n . This paper makes this version explicit for the class of representations which can be realized as representations of the finite group PSL(2,Z/q) for some prime q . The body of the paper is devoted to computing, for the singular representations n , the determinant of the scattering matrix <&{s,n) on which the applications depend. The first application is a version of the Roelcke-Selberg conjecture. This follows from known results once the scattering matrix is given.The study of representations of SL{2, Z) in finite-dimensional vector spaces of (scalar-valued) holomorphic forms dates back to Hecke. Similar problems can be studied for vector spaces of Maass wave forms, with fixed level q and eigenvalue X . One would like to decompose the natural representation of SL(2,Z) in this space, and count the multiplicities of its irreducible components. The eigenvalue estimate obtained for vector-valued forms is equivalent to an asymptotic count, as A -► oo , of these multiplicities.The trace formula for SL(2,Z) can be developed for vector-valued functions which satisfy an automorphic condition involving a group representation n . In fact, Selberg's original paper stated the trace formula in this generality. This paper makes this version explicit for the class of representations which can be realized as representations of the finite group PSL(2,Z/q) for some prime q . We then present some applications. The body of the paper is devoted to computing, for the singular representations n , the determinant of the scattering matrix (s , n) on which the applications depend. The Dirichlet series coefficients of