A Spectral Correspondence for Maaa Waveforms

Let O 1 be a (cocompact) Fuchsian group, given as the group of units of norm one in a maximal order O in an indeenite quaternion division algebra over Q. Using the (classical) Selberg trace formula, we show that the eigenvalues of the automorphic Laplacian for O 1 and their multiplicities coincide with the eigenvalues and multiplicities of the Laplacian deened on the Maaa newforms for the Hecke congruence group ? 0 (d), when d is the discriminant of the maximal order O. We also show the equality of the traces of certain Hecke operators deened on the Laplace eigenspaces for O 1 and the newforms of level d, respectively.