Generation of Siegel modular function fields of even level

For positive integers g and N, let \mathcal {F}_N^{(g)} be the field of meromorphic Siegel modular functions of genus g and level N whose Fourier coefficients belong to the Nth cyclotomic field. We construct explicit generators of \mathcal {F}_N^{(g)} over \mathcal {F}_1^{(g)} by making use of a quotient of theta constants, when g\geq 2 and N is even.