On calculations of the Fourier coefficients of cusp forms of half-integral weight given by the Shintani lift

Hisashi Kojima () and Hiroshi Sakata ()
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Hisashi Kojima: Saitama University
Hiroshi Sakata: Waseda University Senior High School

Indian Journal of Pure and Applied Mathematics, 2025, vol. 56, issue 1, 375-389

Abstract: Abstract Shintani constructed the inverse mapping $$\Psi $$ Ψ of Shimura correspondence $$\Phi $$ Φ from a cusp form F(z) of half-integral weight to the cusp form f(z) of integral weight. The Fourier coefficients of the cusp form $$F_{f}(z)=\Psi (f(z))$$ F f ( z ) = Ψ ( f ( z ) ) are explicitly expressed in terms of periods of a cusp form f(z). Using the reduction theory of integral binary quadratic forms and calculations of periods of f(z), we shall decide an effective algorithm of a calculation of the Fourier coefficients of $$F_{f}(z)$$ F f ( z ) lifted by an cusp form f(z) of small level. Moreover, when f(z) is a cusp form of level 2 and of weight 8, we shall prove that $$F_{f}(z)$$ F f ( z ) is a certain product of some classical theta series of level 4 and of weight 1/2 and certain Dedekind eta functions.

Keywords: Periods of cusp forms; Shintani lift; Modular forms of half-integral weight; 11F67; 11F27; 11F30 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s13226-023-00487-y

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