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Maaß cusp forms for large eigenvalues

Author: Holger Then
Journal: Math. Comp. 74 (2005), 363-381
MSC (2000): Primary 11F72, 11F30; Secondary 11F12, 11Yxx, 11-4, 81Q50
Published electronically: March 23, 2004
MathSciNet review: 2085897
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Abstract: We investigate the numerical computation of Maaß cusp forms for the modular group corresponding to large eigenvalues. We present Fourier coefficients of two cusp forms whose eigenvalues exceed $r=40000$. These eigenvalues are the largest that have so far been found in the case of the modular group. They are larger than the $130$millionth eigenvalue.

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Additional Information

Holger Then
Affiliation: Abteilung Theoretische Physik, Universität Ulm, 89069 Ulm, Germany

Keywords: Automorphic forms, spectral theory, computational number theory, Fourier coefficients, explicit machine computation, multiplicative number theory, Hecke operators, Ramanujan-Petersson conjecture, Sato-Tate conjecture, quantum chaos, Berry conjecture, approximation of special functions, modified Bessel function
Received by editor(s): November 26, 2002
Received by editor(s) in revised form: July 4, 2003
Published electronically: March 23, 2004
Article copyright: © Copyright 2004 American Mathematical Society