TY - JOUR
T1 - On calculations of the Fourier coefficients of cusp forms of half-integral weight given by the Shintani lift
AU - Kojima, Hisashi
AU - Sakata, Hiroshi
N1 - Publisher Copyright:
© 2023, The Indian National Science Academy.
PY - 2023
Y1 - 2023
N2 - Shintani constructed the inverse mapping Ψ of Shimura correspondence 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 Ff(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 Ff(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 Ff(z) is a certain product of some classical theta series of level 4 and of weight 1/2 and certain Dedekind eta functions.
AB - Shintani constructed the inverse mapping Ψ of Shimura correspondence 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 Ff(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 Ff(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 Ff(z) is a certain product of some classical theta series of level 4 and of weight 1/2 and certain Dedekind eta functions.
KW - Modular forms of half-integral weight
KW - Periods of cusp forms
KW - Shintani lift
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U2 - 10.1007/s13226-023-00487-y
DO - 10.1007/s13226-023-00487-y
M3 - Article
AN - SCOPUS:85170247400
SN - 0019-5588
JO - Indian Journal of Pure and Applied Mathematics
JF - Indian Journal of Pure and Applied Mathematics
ER -