Computational aspects of Maass forms for SL(3, Z )

We explore some theoretical and practical aspects of computing L-functions and automorphic forms for the group SL(3, Z ).;The major difficulty in working with automorphic forms is evaluating Jacquet's Whittaker function which occurs in their Fourier expansion. We develop several algorithms for computing this function and study their speed, accuracy, and ranges of applicability. That work culminates in an algorithm for evaluating a given automorphic form at a given point in the generalized upper half-plane.;The other problem that we explore is computing the Dirichlet coefficients of an L-function with a given functional equation. To address that problem, we first use the Dirichelt coefficients to form a Fourier series. Using the functional equations of L-functions twisted by additive characters together with the functional identities of hypergeometric functions, we derive identities satisfied by this Fourier series. One of these new identities serves as a basis of our proposed algorithm for computing the unknown Dirichlet coefficients. The algorithm exploits Hejhal's idea of using the finite Fourier transform and is expected to be more numerically stable than the existing approaches.