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Let Γ be a congruence subgroup of level N in GL n (Z). Let P be a maximal Q -parabolic subgroup of GL n / Q , with unipotent radical U , and let Q = (P ∩ Γ) / (U ∩ Γ). Let p > dim Q ⁡ (U (Q)) + 1 be a prime number that does not divide N. Let M be a (U , p) -admissible Γ-module. Consider the Lyndon-Hochschild-Serre spectral sequence arising from the exact sequence 1 → U ∩ Γ → P ∩ Γ → Q → 1 , which abuts to H ⁎ (P ∩ Γ , M). We show that if M is a trivial U ∩ Γ -module, then certain classes in the E 2 page survive to E ∞. We use this to obtain information about classes in H ⁎ (P ∩ Γ , M) even if M is not a trivial U ∩ Γ -module. This information will be used in future work to prove a Serre-type conjecture for sums of two irreducible Galois representations. [ABSTRACT FROM AUTHOR]