Behavior of theta series of degree $n$ under modular substitutions

Let $F$ be an integral, symmetric, positive definite matrix of order $m\geqslant1$ with an even diagonal. For the theta series of $F$ of degree $n\geqslant1$
$$ \theta_F^{(n)}(Z)=\sum_x^F\exp(\pi i\operatorname{Tr}(^tXFXZ)), $$
where $X$ runs through all integral $m\times n$ matrices and $Z$ is a point of the Siegel upper halfplane of degree $n$, the congruence subgroup of the group $Sp_n(\mathbf Z)$ is found, with respect to which $\theta_F^{(n)}(Z)$ is a Siegel modular form with a multiplicator system (the analog of the group $\Gamma_0(q)$)). The analogous problem is solved for theta series of degree $n$ with spherical functions. The appropriate multiplicator systems are computed for even $m$.
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