  Let $L$ be an integral lattice in the Euclidean space $\mathbb{R}^n$ and $W$
an irreducible representation of the orthogonal group of $\mathbb{R}^n$. We
give an implemented algorithm computing the dimension of the subspace of
invariants in $W$ under the isometry group ${\rm O}(L)$ of $L$. A key step is
the determination of the number of elements in ${\rm O}(L)$ having any given
characteristic polynomial, a datum that we call the {\it characteristic masses}
of $L$. As an application, we determine the characteristic masses of all the
Niemeier lattices, and more generally of any even lattice of determinant $\leq
2$ in dimension $n \leq 25$.
  For Niemeier lattices, as a verification, we provide an alternative (human)
computation of the characteristic masses. The main ingredient is the
determination, for each Niemeier lattice $L$ with non-empty root system $R$, of
the ${\rm G}(R)$-conjugacy classes of the elements of the "umbral" subgroup
${\rm O}(L)/{\rm W}(R)$ of ${\rm G}(R)$, where ${\rm G}(R)$ is the automorphism
group of the Dynkin diagram of $R$, and ${\rm W}(R)$ its Weyl group.
  These results have consequences for the study of the spaces of automorphic
forms of the definite orthogonal groups in $n$ variables over $\mathbb{Q}$. As
an example, we provide concrete dimension formulas in the level $1$ case, as a
function of the weight $W$, up to dimension $n=25$.
