Rationality of generating series for the Fourier coefficient of Siegel modular forms of genus $n$

One proves the rationality of the multiple power series of the form
$$ \sum_{\delta_1\geqslant0}\dots\sum_{\delta_r\geqslant0}a(p_1^{\delta_1}\dots p_r^{\delta_r}N) t_1^{\delta_1}\dots t_r^{\delta_r}, $$
where $a(\dots)$ is the Fourier coefficient of an arbitrary Siegel modular form of genus $n\ge 1$ relative to a congruence subgroup of the group $Sp_n(\mathbf Z)$, $p_1,\dots,p_r$ being a collection of prime numbers, dividing the step of the form.