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2000•Experimental MathematicsOpen access

A Test for Identifying Fourier Coefficients of Automorphic Forms and Application to Kloosterman Sums

Andrew R. Booker

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Abstract

We present a numerical test for determining whether a given set of numbers is the set of Fourier coefficients of a Maass form, without knowing its eigenvalue. Our method extends directly to consideration of holomorphic newforms. The test is applied to show that the Kloosterman sums ±S(l, 1; p)/√p are not the coefficients of a Maass form with small level and eigenvalue. Source code and thecalculated Kloosterman sums are available electron ically.

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We present a numerical test for determining whether a given set of numbers is the set of Fourier coefficients of a Maass form, without knowing its eigenvalue. Our method extends directly to consideration of holomorphic newforms. The test is applied to show that the Kloosterman sums ±S(l, 1; p)/√p are not the coefficients of a Maass form with small level and eigenvalue. Source code and thecalculated Kloosterman sums are available electron ically.

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Available abstract

We present a numerical test for determining whether a given set of numbers is the set of Fourier coefficients of a Maass form, without knowing its eigenvalue. Our method extends directly to consideration of holomorphic newforms. The test is applied to show that the Kloosterman sums ±S(l, 1; p)/√p are not the coefficients of a Maass form with small level and eigenvalue. Source code and thecalculated Kloosterman sums are available electron ically.

Key concepts: Kloosterman sum, Mathematics, Automorphic form, Holomorphic function, Eigenvalues and eigenvectors, Fourier series, Pure mathematics, Set (abstract data type)

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