Maaß cusp forms for large eigenvalues

Abstract

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 130millionth eigenvalue.

Arithmetic Quantum Chaos of Maass Waveforms

“…In order not to miss eigenvalues which lie close together nor to waste CPU time with a too fine grid, we use the adaptive r grid introduced in [38].…”

Computing Maass cusp form on general hyperbolic torus

“…poinst at infinity, and they are formed by quotients of the hyperbolic plane with a discrete subgroup of SL(2,R). It is known that billiards on such punctured surfaces exhibit chaotic behavior and therefore, become a good candidate to the study of quantum chaos [1,2] with applications in cosmology and condensed matter [3]. Different surfaces have been studied for their hyperbolic Laplacian eigenvalues and this includes the modular surface constructed using H/Γ(1) [4] and singly punctured two-torus using H/Γ [5].…”

“…Maass (1949) had systematically studied the real automorphic form, later known as Maass wave forms, and extended Hecke's relation between automorphic forms and Dirichlet series of real analytic automorphic forms [2]. The earliest computation was made numerically in the 70's by Carter(1971) and Maass for r less than 25 [7].…”

“…The earliest computation was made numerically in the 70's by Carter(1971) and Maass for r less than 25 [7]. In 1991, with an enhanced algorithm, Hejhal improved the computation and managed to compute 123 eigenvalues for modular group [2]. Later, Bruggemen (1994) extended the algorithm of Maass waveform to other discrete cofinite subgroups of SL(2, R) [8].…”

Computation of Maass waveforms with nontrivial multiplier systems

“…[27]) which admits generalizations first of all to a remarkably large spectral parameter (cf. [69]), and a further advantage is that it does not depend on any underlying arithmetical properties (i.e. Hecke operators).…”