  Barth and Nieto have found a remarkable quintic threefold which parametrizes
Heisenberg invariant Kummer surfaces which belong to abelian surfaces with a
(1,3)-polarization and a lecel 2 structure. A double cover of this quintic,
which is also a Calabi-Yau variety, is birationally equivalent to the moduli
space {\cal A}_3(2) of abelian surfaces with a (1,3)-polarization and a level 2
structure. As a consequence the corresponding paramodular group \Gamma_3(2) has
a unique cusp form of weight 3. In this paper we find this cusp form which is
\Delta_1^3. The form \Delta_1 is a remarkable weight 1 cusp form with a
character with respect to the paramodular group \Gamma_3. It has several
interesting properties. One is that it admits an infinite product
representation, the other is that it vanishes of order 1 along the diagonal in
Siegel space. In fact \Delta_1 is an element of a short series of modular forms
with this last property. Using the fact that \Delta_1 is a weight 3 cusp form
with respect to the group \Gamma_3(2) we give an independent construction of a
smooth projective Calabi-Yau model of the moduli space {\cal A}_3(2).
