I am seeking a list of automorphism groups of principally polarized abelian varieties in real dimensions 4, 6, 8. These automorphism groups are, of course, related to maximal finite subgroups of the integral symplectic groups $Sp(\mathbb{Z}^4, \omega)$, $Sp(\mathbb{Z}^6, \omega)$, etc.

Some lists and representations are available for dimension 4 (abelian surfaces) in Birkenhake-Lange's book "Complex Abelian Varieties", c.f. $\S 13.4$ and 13.4.4,5,9.

  Reportedly there is some sort of table available for abelian surfaces in Kenji Ueno's paper ``On fibre spaces of normally polarized abelian varieties of dimension 2", J. Fac. Science, Univ. Tokyo, Sect. IA 18 37-95 1971 . However I cannot locate Ueno's paper online.

My question: can anybody provide an easy-to-read table describing the linear representations of the automorphism groups of abelian varieties in low dimensions?

For instance, a simple matrix representation of some maximal finite subgroups of $Sp(\mathbb{Z}^4, \omega)$, $Sp(\mathbb{Z}^6, \omega)$ would be much desired.

(**%Edited after abx comment%**)

I seek explicit linear integral representations $\rho: Aut(X,\omega) \to Sp_{2g}(\mathbb{Z})$, when $(X,\omega)$ is complex $g$-dimensional PPAV. I prefer explicit matrix representations because I want to compute some orbits on vector spaces.

A list of interesting automorphism groups (with integral symplectic representations) of principally polarized abelian varieties in real dimensions 4, 6, 8 would be useful. These automorphism groups are, of course, related to maximal finite subgroups of the integral symplectic groups $Sp(\mathbb{Z}^4, \omega)$, $Sp(\mathbb{Z}^6, \omega)$, etc.

Some lists and representations are available for dimension 2 (abelian surfaces) in Birkenhake-Lange's book "Complex Abelian Varieties", c.f. $\S 13.4$ and 13.4.4,5,9. Reportedly there is some sort of table available for abelian surfaces in Kenji Ueno's paper ``On fibre spaces of normally polarized abelian varieties of dimension 2", J. Fac. Science, Univ. Tokyo, Sect. IA 18 37-95 1971 . However I cannot locate Ueno's paper online.

The comment of @abx below is useful: in low dimensions, PPAVs are Jacobians of Riemann surfaces:

For complex surfaces, the Bolza surface is the maximally symmetric genus 2 Riemann surface with automorphism group $\approx PGL(2,3)$. Here my question amounts to:

Question: can anybody specify an explicit faithful linear representation of $PGL(2,3) \to Sp(\mathbb{Z}^4, \omega)$ ?

For complex dimension 3, the Appendix 1 of Conway/Sloane in Buser/Sarnak's "On the period matrix of a Riemann surface of large genus" is useful: the automorphism group of the Barnes-Wall lattice $A_6 ^{(2)}$ is the ``most interesting" automorphism group, and my question amounts to:

**Question: Can anybody specify an explicit faithful linear representation of $Aut(A_6 ^{(2)}) \to Sp(\mathbb{Z}^6, \omega)$** ?

For complex dimension 4, the $E_8$ lattice has the largest automorphism group, and here our question amounts to:

**Question: Can anybody specify an explicit faithful linear representation of $Aut(E_8) \to Sp(\mathbb{Z}^8, \omega)$** ?

I am searching through the various Atlas/GAP/MAGMA/SAGE databases on finite group representations, but have not yet located the desired representations.