Generators of Siegel modular function field of higher genus and level

For positive integers $g$ and $N$, let ${\mathcal{F}}_N$ be the field of meromorphic Siegel modular functions of genus $g$ and level $N$ whose Fourier coefficients belong to the $N$th cyclotomic field. We present explicit generators of ${\mathcal{F}}_N$ over ${\mathcal{F}}_1$ in terms of quotient of theta constants, when $g\ge 2$ and $N\ge 3$.