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

Shintani constructed the inverse mapping Psi of Shimura correspondence Phi 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)) 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) 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) is a certain product of some classical theta series of level 4 and of weight 1/2 and certain Dedekind eta functions.