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
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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
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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 $\Gamma\_2[2]$.



Reference on elements of finite order in principal congruence subgroups of symplectic groups
This is a well known argument (for any even $g \geq 2$, and for other integral matrix groups) . If $x$ is an element in the level $q$ congruence subgroup, then $\frac{x-I}{q}$ is an integral matrix, so has all its eigenvalues algebraic integers. If $\alpha$ is any eigenvalue of $x,$ then $\alpha$ is a root of unity as $x$ has finite order. On the other hand, $\frac{\alpha - 1}{q}$ is also an algebraic integer, and note that its absolute value is less than $1$ as $q \geq 3$. The same applies to any of its algebraic conjugates. As was known to W. Burnside, the only algebraic integer with all algebraic conjugates of absolute value less than $1$ 
is $0$ ( for otherwise, the geometric mean of their absolute values would be at least $1$, hence the arithmetic mean of those eigenvalues would greater than $1$, which is not the case).
Hence all eigenvalues of $x$ are $1$, and $x =I,$ as $x$ has finite order.


