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We describe a method to compute mod    $$\ell $$  ℓ Galois representations contained in the    $${{\text {H}}}_{\acute{\mathrm{e}}\mathrm{t}}^2$$  H e ´ t 2 of surfaces. We apply this method to the case of a representation with values in    $${\text {GL}}_3(\mathbb {F}_9)$$  GL 3 ( F 9 ) attached to an eigenform over a congruence subgroup of    $${\text {SL}}_3$$  SL 3 . We obtain, in particular, a polynomial with Galois group isomorphic to the simple group    $${\text {PSU}}_3(\mathbb {F}_9)$$  PSU 3 ( F 9 ) and ramified at 2 and 3 only.

foundations of computational mathematics

Explicit Computation of a Galois Representation Attached to an Eigenform Over $${\text {SL}}_3$$ from the $${{\text {H}}}_{\acute{\mathrm{e}}\mathrm{t}}^2$$ of a Surface