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

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.