We study the appearances of \mathsf{LS}_{12} and \mathsf{S}_{16} in the weight-graded compactly supported cohomology of moduli spaces of curves. As applications, we prove new nonvanishing results for the middle cohomology groups of \mathcal{M}_9 and \mathcal{M}_{11} and give evidence to support the conjecture that the dimension fo H^{2g + k}_c(\mathcal{M}_g) grows at least exponentially with g for almost all k.
