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Computation of Maass cusp forms on modular group in Mathematica

Abubaker A. Siddig, Hishamuddin Zainuddin

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Abstract

Maass cusp forms of the Fuchsian group are eigenfunctions of the hyperbolic Laplacian representing bound states. It is known that the Maass cusp forms can only be solved numerically. The aim of this paper is to ex-tend Hejhal's algorithm to compute Maass cusp form on the modular group SL(2, R)\H using Mathematica with the intent of exploiting its graphical and symbolic capabilities. We compute the eigenvalues of the Maass cusp forms em-ploying a graphical scheme to locate the approximate value of the eigenvalue. We also plotted Maass cusp forms corresponding to selected eigenvalues.

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What this paper is about

Maass cusp forms of the Fuchsian group are eigenfunctions of the hyperbolic Laplacian representing bound states. It is known that the Maass cusp forms can only be solved numerically. The aim of this paper is to ex-tend Hejhal's algorithm to compute Maass cusp form on the modular group SL(2, R)\H using Mathematica with the intent of exploiting its graphical and symbolic capabilities. We compute the eigenvalues of the Maass cusp forms em-ploying a graphical scheme to locate the approximate value of the eigenvalue. We also plotted Maass cusp forms corresponding to selected eigenvalues.

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Available abstract

Maass cusp forms of the Fuchsian group are eigenfunctions of the hyperbolic Laplacian representing bound states. It is known that the Maass cusp forms can only be solved numerically. The aim of this paper is to ex-tend Hejhal's algorithm to compute Maass cusp form on the modular group SL(2, R)\H using Mathematica with the intent of exploiting its graphical and symbolic capabilities. We compute the eigenvalues of the Maass cusp forms em-ploying a graphical scheme to locate the approximate value of the eigenvalue. We also plotted Maass cusp forms corresponding to selected eigenvalues.

Key concepts: Cusp (singularity), Cusp form, Modular group, Mathematics, Eigenvalues and eigenvectors, Group (periodic table), Fuchsian group, Modular form

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