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A Note On Signs Of Fourier Coefficients Of Two Cusp Forms

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Soumyarup Banerjee

Number Theory · Submitted 26 Sep 2016

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim In this note we compare the signs of the Fourier coefficients of two cusp forms simultaneously for the congruence subgroup $\Gamma_0(\mathit{N})$ where the coefficients lie in an arithmetic progression.

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arXiv ID1609.07938 (version 2)
Pith IDUJ2Z6TAK, Pith's permanent record of this paper and its review
CitationSoumyarup Banerjee. "A Note On Signs Of Fourier Coefficients Of Two Cusp Forms." arXiv:1609.07938 (2016). https://pith.science/paper/UJ2Z6TAK
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,326 input tokens.

What the abstract itself states, as probabilities: claims first 0.07 · states limitation 0.13 · human participants 0.01 · controlled comparison 0.04 · randomized assignment 0.01 · states quantitative result 0.06 · compares with prior results 0.93 · mentions code or data availability 0.01.

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arXiv listing abstract v2. Results and the main claim are underlined; .

Kohnen and Sengupta proved that two cusp forms of different integral weights with real algebraic Fourier coefficients have infinitely many Fourier coefficients of the same as well as of opposite sign, up to the action of a Galois automorphism. Recently Gun, Kohnen and Rath strengthen their result by comparing the simultaneous sign changes of Fourier coefficients of two cusp forms with arbitrary real Fourier coefficients. The simultaneous sign changes of Fourier coefficients of two same integral weight cusp forms follow from an earlier work of Ram Murty. In this note we compare the signs of the Fourier coefficients of two cusp forms simultaneously for the congruence subgroup $\Gamma_0(\mathit{N})$ where the coefficients lie in an arithmetic progression. Next we consider an analogous question for the particular sparse sequences of Fourier coefficients of normalized Hecke eigen cusp forms for the full modular group.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 Kohnen and Sengupta proved that two cusp forms of different integral weights with real algebraic Fourier coefficients have infinitely many Fourier coefficients of the same as well as of opposite sign, up to the action of a Galois automorphism. background 1.00 0.03 0.06 asserted 1.00
s1 Recently Gun, Kohnen and Rath strengthen their result by comparing the simultaneous sign changes of Fourier coefficients of two cusp forms with arbitrary real Fourier coefficients. background 1.00 0.07 0.06 asserted 1.00
s2 The simultaneous sign changes of Fourier coefficients of two same integral weight cusp forms follow from an earlier work of Ram Murty. background 0.99 0.03 0.04 asserted 1.00
s3 In this note we compare the signs of the Fourier coefficients of two cusp forms simultaneously for the congruence subgroup $\Gamma_0(\mathit{N})$ where the coefficients lie in an arithmetic progression. contribution 1.00 0.91 0.25 asserted 0.83
s4 Next we consider an analogous question for the particular sparse sequences of Fourier coefficients of normalized Hecke eigen cusp forms for the full modular group. contribution 0.92 0.72 0.06 unclear 0.81

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Original abstract

Kohnen and Sengupta proved that two cusp forms of different integral weights with real algebraic Fourier coefficients have infinitely many Fourier coefficients of the same as well as of opposite sign, up to the action of a Galois automorphism. Recently Gun, Kohnen and Rath strengthen their result by comparing the simultaneous sign changes of Fourier coefficients of two cusp forms with arbitrary real Fourier coefficients. The simultaneous sign changes of Fourier coefficients of two same integral weight cusp forms follow from an earlier work of Ram Murty. In this note we compare the signs of the Fourier coefficients of two cusp forms simultaneously for the congruence subgroup $\Gamma_0(\mathit{N})$ where the coefficients lie in an arithmetic progression. Next we consider an analogous question for the particular sparse sequences of Fourier coefficients of normalized Hecke eigen cusp forms for the full modular group.

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