  We derive a mass formula for n-dimensional unimodular lattices having any
prescribed root system. We use Katsurada's formula for the Fourier coefficients
of Siegel Eisenstein series to compute these masses for all root systems of
even unimodular 32-dimensional lattices and odd unimodular lattices of
dimension n < 31. In particular, we find the mass of even unimodular
32-dimensional lattices with no roots, and the mass of odd unimodular lattices
with no roots in dimension n < 31, verifying Bacher and Venkov's enumerations
in dimensions 27 and 28. We also compute better lower bounds on the number of
inequivalent unimodular lattices in dimensions 26 to 30 than those afforded by
the Minkowski-Siegel mass constants.
  The ASCII text file table.txt (1.3 MB) at arXiv:math.NT/0012231 includes the
complete table of masses of 32-dimensional even unimodular lattices with any
given root system, and may be dowloaded using the source link.
