Effective computation of Maass cusp forms

We study theoretical and practical aspects of high-precision computation of Maass forms. First, we compute to over 1000 decimal places the Laplacian and Hecke eigenval-ues for the first few Maass forms on PSL(2,Z)\. Second,we give an algorithm for rigor-ously verifying that a proposed eigenvalue together with a proposed set of Fourier coeffi-cients indeed correspond to a true Maass cusp form. We apply this to prove that our val-ues for the first ten eigenvalues on PSL(2,Z)\ are correct to at least 100 decimal places. Third, we test some algebraicity properties of the coefficients, among other things giv-ing evidence that the Laplacian and Hecke eigenvalues of Maass forms on PSL(2,Z)\ are transcendental. 1