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

• 11F46 - Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms
• 11F80 - Galois representations
• 14H10 - Families, moduli of curves (algebraic)
• 14K10 - Algebraic moduli of abelian varieties, classification