A spectral correspondence for Maaß waveforms

Let O1 be a (cocompact) Fuchsian group, given as the group of units of norm one in a maximal order O in an indefinite quaternion division algebra over Q. Using the (classical) Selberg trace formula, we show that the eigenvalues of the automorphic Laplacian for O1 and their multiplicities coincide with the eigenvalues and multiplicities of the Laplacian defined on the Maaß newforms for the Hecke congruence group Γ0(d), when d is the discriminant of the maximal order O. We also show the equality of the traces of certain Hecke operators defined on the Laplace eigenspaces for O1 and the newforms of level d, respectively.