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.

中文翻译：

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SL4(Z) 和伽罗瓦表示的同余子群的扭曲一维系数的上同调

我们扩展 [AGM1, AGM2, AGM3] 中的计算以找到 SL(4,Z) 的同余子群 Gamma 的 5 次上同调，其系数在域 K 中，由 nebentype 字符 eta 扭曲，以及赫克代数。这是最高尖度。在实践中，我们取 K 为一个大特征的有限域，作为复数的代理。对于找到的每个 Hecke 特征类，我们生成一个似乎附加到它的伽罗瓦表示。我们的计算表明，在每种情况下，这种伽罗瓦表示都是唯一可以附加到它的表示。所附伽罗瓦表示的存在与 Scholze 定理一致，并阐明了 Gamma 的 Borel-Serre 边界。计算需要对我们以前的算法进行重大修改以适应扭曲系数。非平凡系数给我们的数据结构增加了一层复杂性，并且出现了必须在 Galois Finder 中考虑的新可能性，该代码用于查找 Galois 表示。我们改进了 Galois Finder，以便它报告附加的 Galois 表示何时由我们的数据唯一确定。