The authors determine the number of level $1$, polarized, algebraic regular, cuspidal automorphic representations of $mathrm_n$ over $mathbb Q$ of any given infinitesimal character, for essentially all $n leq 8$. For this, they compute the dimensions of spaces of level $1$ automorphic forms for certain semisimple $mathbb Z$-forms of the compact groups $mathrm_7$, $mathrm_8$, $mathrm_9$ (and ${mathrm G}_2$) and determine Arthur's endoscopic partition of these spaces in all cases. They also give applications to the $121$ even lattices of rank $25$ and determinant $2$ found by...