Computation of Maass waveforms with nontrivial multiplier systems

Strömberg, Fredrik

doi:10.1090/s0025-5718-08-02129-7

Cited by 15 publications

(12 citation statements)

References 53 publications

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“…The algorithm which we use to compute Maass forms and Eisenstein series was originally developed by Dennis A. Hejhal (based on an idea of Harold Stark) for computing Maass cusp forms on Hecke triangle groups [22,23]. Subsequently this algorithm was adapted by the author to other Fuchsian groups, non-trivial weights and multiplier systems and even harmonic weak Maass forms [12,64,65]. It was also generalized by Helen Avelin to compute Eisenstein series and Green's functions [5,6].…”

Section: On the Computation Of Maass Forms For General Subgroupsmentioning

confidence: 99%

Noncongruence subgroups and Maass waveforms

Strömberg

2019

Journal of Number Theory

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The main topic of the paper is spectral theory for noncongruence subgroups of the modular group. We have studied a selection of the main conjectures in the area: the Roelcke-Selberg and Phillips-Sarnak conjectures and Selberg's conjecture on exceptional eigenvalues. The first two concern the existence and nonexistence of an infinite discrete spectrum for certain types of Fuchsian groups and last states that there are no exceptional eigenvalues for congruence subgroups, or in other words, there is a specific spectral gap in the cuspidal spectrum.Our main theoretical result states that if the corresponding surface has a reflectional symmetry which preserves the cusps then the Laplacian on this surface has an infinite number of "new" discrete eigenvalues. We define old and new spaces of Maass cusp forms for noncongruence subgroups in a way that provides a natural generalization of the usual definition from congruence subgroups and give a method for determining the structure of the old space algorithmically.In addition to the theoretical result we also present computational data, including a table of subgroups of the modular group and eigenvalues of Maass forms for noncongruence subgroups. We also present, for the first time, numerical examples of both exceptional and (non-trivial) residual eigenvalues. To be able to (even heuristically) cerify lists of computed eigenvalues we also proved an explicit average Weyl's law in this setting.

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Section: On the Computation Of Maass Forms For General Subgroupsmentioning

confidence: 99%

Noncongruence subgroups and Maass waveforms

Strömberg

2019

Journal of Number Theory

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“…It is interesting to evaluate numerically some examples of Theorem 4. This is possible thanks to computations done by Strömberg [61]. Note that half-integral weight Fourier coefficients, even in the holomorphic case, are notoriously difficult to compute.…”

Section: Weyl's Law Gives That Asmentioning

confidence: 99%

Geometric invariants for real quadratic fields

2016

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Abstract. To an ideal class of a real quadratic field we associate a certain surface. This surface, which is a new geometric invariant, has the usual modular closed geodesic as its boundary. Furthermore, its area is determined by the length of an associated backward continued fraction. We study the distribution properties of this surface on average over a genus. In the process we give an extension and refinement of the Katok-Sarnak formula.

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“…Except for a thin set arising from quadratic fields, there is no known explicit construction of these functions. Thus, for most Maass forms one must rely on computer calculations to determine numerical approximations [12,15,28]. Recently, such computations have been proven to be correct [5].…”

Section: Prior Methodsmentioning

confidence: 99%

Maass Forms on GL(3) and GL(4)

Farmer

Koutsoliotas

Lemurell

2013

International Mathematics Research Notices

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We describe a practical method for finding an L−function without first finding the associated underlying object. The procedure involves using the Euler product and the approximate functional equation in a new way. No use is made of the functional equation of twists of the L−function. The method is used to find a large number of Maass forms on SL(3, Z) and to give the first examples of Maass forms of higher level on GL(3), and on GL(4) and Sp(4).

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Computation of Maass waveforms with nontrivial multiplier systems

Cited by 15 publications

References 53 publications

Noncongruence subgroups and Maass waveforms

Noncongruence subgroups and Maass waveforms

Geometric invariants for real quadratic fields

Maass Forms on GL(3) and GL(4)

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