  For two varieties of dimension $5$ constructed as double covers of ${\mathbb
P}^5$ branched along the union of $12$ hyperplanes, we prove that the number of
points over ${\mathbb F}_p$ can be expressed in terms of Artin symbols and the
$p$th Fourier coefficients of modular forms. Many analogous results are known
in dimension $\le 3$, but very few in higher dimension. In addition, we use an
idea of Burek to construct quotients of our varieties for which the point
counts mod $p$ are expressible in terms of Artin symbols and the coefficients
of a single modular form of weight $6$.
