  We extend the computations in [AGM1, AGM2, AGM3] to find the cohomology in
degree five of a congruence subgroup Gamma of SL(4,Z) with coefficients in a
field K, twisted by a nebentype character eta, along with the action of the
Hecke algebra. This is the top cuspidal degree. In practice we take K to be a
finite field of large characteristic, as a proxy for the complex numbers. For
each Hecke eigenclass found, we produce a Galois representation that appears to
be attached to it. Our computations show that in every case this Galois
representation is the only one that could be attached to it. The existence of
the attached Galois representations agrees with a theorem of Scholze and sheds
light on the Borel-Serre boundary for Gamma.
  The computations require serious modifications to our previous algorithms to
accommodate the twisted coefficients. Nontrivial coefficients add a layer of
complication to our data structures, and new possibilites arise that must be
taken into account in the Galois Finder, the code that finds the Galois
representations. We have improved the Galois Finder so that it reports when the
attached Galois representation is uniquely determined by our data.
