Effective computation of Maass cusp forms

Author please provide the abstract. Please provide the abstract of this paper that should not exceed 150 words (including spaces) and citation free. 1 Preliminary The aim of this paper is to address theoretical and practical aspects of high-precision computation of Maass forms. Namely, we compute to over 1000 decimal places the Laplacian and Hecke eigenvalues for the first few Maass forms on PSL(2, Z)\H, and certify the Laplacian eigenvalues correct to 100 places. We then use these computations to test certain algebraicity properties of the coefficients. The outline of the paper is as follows. In Section 2, we discuss Hejhal’s algorithm for computation of Maass forms on cofinite Fuchsian groups with cusps, and the details necessary to implement it in high precision. This algorithm is heuristic and does not prove the existence of cusp forms. In Section 3 we turn to the question of rigorously verifying that a proposed eigenvalue, together with a proposed set of Fourier coefficients, indeed correspond to a true Maass cusp form. We will use standard methods to show that the putative eigenfunction has almost all of its spectral support concentrated near the proposed eigenvalue. It is a more subtle point to show that it is close to a cusp form