of the Laplacian; in other words Using the variable substitution the eigenvalues are labeled Then the Selberg trace formula is given by The right hand side is a sum over conjugacy classes of the group , with the first term corresponding to the identity element and the remaining terms forming a sum over the other conjugacy classes (which are all hyperbolic in this case). The function has to satisfy the following: be analytic on ; ; there exist positive constants and such that: The function is the Fourier transform of , that is, Notes References External links Selberg trace formula resource page Automorphic forms