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We consider the problem of explicitly computing dimensions of spaces ofautomorphic or modular forms in level one, for a split classical group$\mathbf{G}$ over $\mathbb{Q}$ such that $\mathbf{G}(\R)$ has discrete series.Our main contribution is an algorithm calculating orbital integrals for thecharacteristic function of $\mathbf{G}(\mathbb{Z}_p)$ at torsion elements of$\mathbf{G}(\mathbb{Q}_p)$. We apply it to compute the geometric side inArthur's specialisation of his invariant trace formula involving stablediscrete series pseudo-coefficients for $\mathbf{G}(\mathbb{R})$. Therefore weexplicitly compute the Euler-Poincar\'e characteristic of the level onediscrete automorphic spectrum of $\mathbf{G}$ with respect to afinite-dimensional representation of $\mathbf{G}(\mathbb{R})$. For such a group$\mathbf{G}$, Arthur's endoscopic classification of the discrete spectrumallows to analyse precisely this Euler-Poincar\'e characteristic. For exampleone can deduce the number of everywhere unramified automorphic representations$\pi$ of $\mathbf{G}$ such that $\pi_{\infty}$ is isomorphic to a givendiscrete series representation of $\mathbf{G}(\mathbb{R})$. Dimension formulaefor the spaces of vector-valued Siegel modular forms are easily derived.

Dimensions of spaces of level one automorphic forms for split classical groups using the trace formula