  Let $I$ be an $\m$-primary ideal of a Noetherian local ring $(R, \m)$ of
positive dimension. The coefficient $e_1(\mathcal I)$ of the Hilbert polynomial
of an $I$-admissible filtration $\mathcal I$ is called the Chern number of
$\mathcal I$. A formula for the Chern number has been derived involving Euler
characteristic of subcomplexes of a Koszul complex. Specific formulas for the
Chern number have been given in local rings of dimension at most two. These
have been used to provide new and unified proofs of several results about
$e_1(\mathcal I)$.
