Cohomology of moduli spaces via a result of Chenevier and Lannes

We use a classification result of Chenevier and Lannes for algebraic automorphic representations together with a conjectural correspondence with $\mathrm{\ell }$-adic absolute Galois representations to determine the Euler characteristics (with values in the Grothendieck group of such representations) of ${\bar{\mathcal{M}}}_{3,n}$ and ${\mathcal{M}}_{3,n}$ for $n\le 14$ and of local systems ${\mathbb{V}}_{\lambda }$ on ${\mathcal{A}}_3$ for $\mathrm{\mid }\lambda \mathrm{\mid }\le 16$.