We should start with the definition of the symplectic group for an arbitrary ring $R$. The symplectic group $Sp(g,R)$ is the subgroup of $SL(2g,R)$ such that all elements satisfy $M=J_g^t M J_g$ with $J_g$ being the canonical almost complex structure - or involution whatever you prefer to call it.

The principal congruence subgroup of level q is defined as $$ \Gamma_g[q]:= ker\left(Sp(g,\mathbb{Z})\to Sp(g,\mathbb{Z}/q\mathbb{Z})\right).$$

## The actual question

I am searching for a reference that the only element of finite order in $\Gamma_2[q]$ for $q \geq 3$ is the identity matrix. Or expressed in a formula : $$\forall\ q \geq 3 \quad\forall\ M \in \Gamma_2[q] : \quad M^n=I \Longrightarrow M=I .$$

## The application

The above result implies that elements of finite order in $\Gamma_2[2]$ are of order 2. A reference on this would also be appreciated. This allows me to apply a theorem on involutions on matrices of finite order in $\Gamma_2[2]$.