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.