  In this article, we determine the trace of some Hecke operators on the spaces
of level one automorphic forms on the special orthogonal groups of the
euclidean lattices $\mathrm{E}_7$, $\mathrm{E}_8$ and $\mathrm{E}_8\oplus
\mathrm{A}_1$, with arbitrary weight. Using Arthur's theory, we deduce
properties of the Satake parameters of the automorphic representations for the
linear groups discovered by Chenevier and Renard. Our results corroborate a
conjecture by Bergstr\"om, Faber and van der Geer about the Hasse-Weil zeta
function on the moduli spaces of $17$-pointed curves of genus $3$.
