We generate extensions of $\Q$ with Galois group $\SL_3(\F_2)$ giving rise to three-dimensional mod 2 Galois representations with sufficiently low level to allow the computational testing of a conjecture of Ash, Doud, Pollack, and Sinnott relating such representations to mod 2 arithmetic cohomology. We test the conjecture for these examples and offer a refinement of the conjecture that resolves ambiguities in the predicted weight.
"$\SL_3(\F_2)$-Extensions of $\Q$ and Arithmetic Cohomology Modulo 2." Experiment. Math. 13 (3) 297 - 307, 2004.