Symplectic group of degree four

Definition
Let $$K$$ be a field. The group $$Sp(4,K)$$ is defined as the symplectic group of degree four over $$K$$. Explicitly, it is the group given as:

$$\{ A \in GL(4,K) \mid A \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\\end{pmatrix}A^T = \begin{pmatrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\\end{pmatrix} \}$$

We could give an alternative definition, which yields a different but conjugate subgroup of $$GL(4,K)$$ (and hence the same group up to isomorphism):

$$\{ A \in GL(4,K) \mid A \begin{pmatrix} 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \\\end{pmatrix}A^T = \begin{pmatrix} 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \\\end{pmatrix} \}$$

For $$q$$ a prime power, we denote by $$Sp(4,q)$$ the group $$Sp(4,\mathbb{F}_q)$$ where $$\mathbb{F}_q$$ is the (unique up to isomorphism) field of size $$q$$.

Arithmetic functions
We list here the arithmetic functions for $$Sp(4,q)$$ for a prime power $$q$$.