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We sketch a method to compute mod $\ell$ Galois representations contained inthe H2 \'etale of surfaces. We apply this method to the case of arepresentation with values in GL(3,9) attached to an eigenform over acongruence subgroup of SL(3). We obtain in particular a polynomial with Galoisgroup isomorphic to the simple group PSU(3,9) and ramified at 2 and 3 only.

Explicit computation of a Galois representation attached to an eigenform over SL3 from the H2 \'etale of a surface