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.

A generic element of $Sp(g,R)$ is denoted by $M=(A B \\ C D)$ where $A,B,C,D$ are $g\times g$ matrices.

Note bene, if $R$ is Euclidean then $Sp(g,R)$ is generated by the involution $J_g$ and the translations $\begin{pmatrix}I_g\ S \\ 0_g \ I_g \end{pmatrix}$ sending $Z$ to $Z+S$ where $S$ is a symmetric $g \times g$ matrix.

## Why am I interested in these generators ?

Well, first of all I am interested in modular forms. They are holomorphic functions $f:\mathbb{H}_g \to V $ transforming under a subgroup $\Gamma$ of a symplectic group as follows $$f(M \cdot Z)=j(M,Z)\cdot f(Z)\quad \quad \forall\ M \in \Gamma ,$$ where $j$ is a factor of automorphy. This means that $j: \Gamma \times \mathbb{H}_g \to GL(V)$ is holomorphic in the second variable and satisfies the cocycle relation $$j(MN,Z)=j(M,N \cdot Z) \cdot j(N,Z).$$

Hence, it suffices to check the first equation only for the generators of $\Gamma$.

Sometimes we have modular forms to a proper subgroup and even know how they transform under the full symplectic group. But they do not transform with a factor of automorphy. Examples would be the theta series $$f_a(Z):=\sum_{\nu \in \mathbb{Z}^g}{exp \left(2\pi i \left(\nu+\frac{a}{2}\right)^t Z \left(\nu+\frac{a}{2}\right) \right)}, \quad \quad a \in \mathbb{F}_2^g$$ They are modular forms to certain proper subgroups. But the action of $Sp(g,\mathbb{Z})$'s generators can be given quite easily, roughly speaking : the translations scale the thetas and involution returns a linear combination of all thetas. That way, it is possible to determine whether a polynomial in the different theta series is a modular form to the full modular group.

## The actual question

I would be very pleased if someone could give a reference for the generators of subgroups of $Sp(g,\mathbb{Z})=\Gamma_g$ like $$ \Gamma_g[q]:= ker\left(Sp(g,\mathbb{Z})\to Sp(g,\mathbb{Z}/q\mathbb{Z})\right)$$ $$ \Gamma_{g,0}[N]:=\left\{ M \in \Gamma_g : C \equiv 0 \mod N \right\} $$ $$ \Gamma_{g}^{0}[N]:=\left\{ M \in \Gamma_g : B \equiv 0 \mod N \right\} $$ and others if you know them, too. In particular, I am interested in $g=2$. I guess this way it is faster and I cannot make any mistakes. As hinted above it would be also nice if these generators could be given in terms of the generators of $Sp(g,\mathbb{Z})$.

Thanks Tom

p.s. it would be nice if someone could help me fixing the brackets in the above definition of $\Gamma_{g,0}[N]$.

edit1 : added notation $M=(A B \\ C D)$