  The aim of this paper is twofold. First, we introduce a new method for
evaluating the multiplicity of a given discrete series in the space of level
$1$ automorphic forms of a split classical group $G$ over $\mathbb{Z}$, and
provide numerical applications in absolute rank $\leq 8$. Second, we prove a
classification result for the level one cuspidal algebraic automorphic
representations of ${\rm GL}_n$ over $\mathbb{Q}$ ($n$ arbitrary) whose motivic
weight is $\leq 24$.
  In both cases, a key ingredient is a classical method based on the Weil
explicit formula, which allows to disprove the existence of certain level one
algebraic cusp forms on ${\rm GL}_n$, and that we push further on in this
paper. We use these vanishing results to obtain an arguably ``effortless''
computation of the elliptic part of the geometric side of the trace formula of
$G$, for an appropriate test function.
  Thoses results have consequences for the computation of the dimension of the
spaces of (possibly vector-valued) Siegel modular cuspforms for ${\rm
Sp}_{2g}(\mathbb{Z})$: we recover all the previously known cases without
relying on any, and go further, by a unified and ``effortless'' method.
