Consider the congruence subgroup $\Gamma_g(n)=\left\{\begin{bmatrix}a&b\\c&d\end{bmatrix}\in Sp_{2g}(\mathbb{Z}):\begin{bmatrix}a&b\\c&d\end{bmatrix}\equiv\begin{bmatrix}1_g&0\\0&1_g\end{bmatrix}\pmod{n} \right\}$. The moduli space of principally polarized abelian varieties with a choice of symplectic basis for the $n$-torsion subgroup $A[n]$ is given by $\mathbb{H}_g/\Gamma_g(n)$.
The congruence subgroup $\Gamma_g(n,2n)=\left\{\begin{bmatrix}a&b\\c&d\end{bmatrix}\in \Gamma_g(l): diag(ab^t)=diag(cd^t)\equiv 0\pmod{2n}\right\}$ supposedly gives us the moduli space of p.p.a.v. with a choice of symplectic basis and a choice of $2n$-torsion point via the quotient $\mathbb{H}_g/\Gamma_g(n)$ (see Grushevsky's paper on Schottky problem).
How do we interpret the conditions $diag(ab^t)=diag(cd^t)\equiv 0\pmod{2n}$ in terms of the moduli problem?
I know how the first congruence subgroup works from congruence subgroups of $SL_2(\mathbb{Z})$ and how it works for $Sp_{2g}(\mathbb{Z})$. For $SL_2(\mathbb{Z})$, we have three kinds of notable congruence subgroups, $\Gamma_0(n), \Gamma_1(n), \Gamma(n)$. The modular curves, $Y_0(n), Y_1(n), Y(n)$ describe the moduli space of elliptic curves with a choice of cyclic $n$-torion subgroup, moduli space of elliptic curves with a choice of $n$-torsion point and moduli space of elliptic curves with a choice of symplectic basis on $E[n]$.
I wanted to try and understand how the congruence subgroup $\Gamma_g(n,2n)$ for the simplest case $g=1$. Here I am having some doubt.
For $g=1$, $\Gamma_g(n,2n)$ is the set of matrices in $SL_2(\mathbb{Z})$ is given by the conditions $b\equiv c\equiv 0\pmod{2n}$ and $a\equiv d\equiv 1\pmod{n}$.
If I want to look at the elliptic curves with symplectic basis for $E[n]$ and a $2n$-torsion point, the corresponding congruence subgroup ought to be $\Gamma(n)\cap \Gamma_1(n)\subset SL_2(\mathbb{Z})$. But I don't think this agrees with $\Gamma_1(n,2n)$.
Why doesn't $\mathbb{H}/(\Gamma(n)\cap \Gamma_1(n))$ become the moduli space of elliptic curves with a choice of sympletic basis for $E[n]$ and a choice of $2n$-torsion point?