A spectral correspondence for Maa beta waveforms

Let O-1 be a (cocompact) Fuchsian group, given as the group of units of nor m one in a maximal order O in an indefinite quaternion division algebra ove r Q. Using the (classical) Selberg trace formula, we show that the eigenval ues of the automorphic Laplacian for O-1 and their multiplicities coincide with the eigenvalues and multiplicities of the Laplacian defined on the Maa beta newforms for the Hecke congruence group Gamma(0)(d), when d is the di scriminant of the maximal order O. We also show the equality of the traces of certain Hecke operators defined on the Laplace eigenspaces for O-1 and t he newforms of level d, respectively.