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 eigenvalues for the first few Maass forms on PSL(2, Z)\H. Second, we give an algorithm for rigorously verifying that a proposed eigenvalue together with a proposed set of Fourier coefficients indeed correspond to a true Maass cusp form. We apply this to prove that our values for the first ten eigenvalues on PSL( 2, Z)\H are correct to at least 100 decimal places. Third, we test some algebraicity properties of the coefficients, among other things giving evidence that the Laplacian and Hecke eigenvalues of Maass forms on PSL( 2,Z)\H are transcendental.