Abstract: In two previous papers \cite{AGM1, AGM2} we computed cohomology groups $H^{5}(\Gamma_{0} (N); \C)$ for a range of levels~$N$, where $\Gamma_{0} (N)$ is the congruence subgroup of $\SL_{4} (\Z)$ consisting of all matrices with bottom row congruent to $(0,0,0,*)$ mod~$N$. In this note we update this earlier work by carrying it out for prime levels up to $N = 211$. This requires new methods in sparse matrix reduction, which are the main focus of the paper. Our computations involve matrices with up to 20~million non-zero entries. We also make two conjectures concerning the contributions to $H^{5}(\Gamma_{0} (N); \C)$ for $N $ prime coming from Eisenstein series and Siegel modular forms.