  We study a natural Hodge module on the Hilbert scheme of four points on
affine three-space, which categorifies the Donaldson--Thomas invariant of the
Hilbert scheme. We determine the weight filtration on the Hodge module
explicitly in terms of intersection cohomology complexes, and compute the
E-polynomial of its cohomology. The computations make essential use of a
description of the singularity of the Hilbert scheme as the degeneracy locus of
the Pfaffian function.
