  The main goal of this paper is to characterize the Hilbert functions of all
(artinian) codimension 4 Gorenstein algebras that have at least two independent
relations of degree four. This includes all codimension 4 Gorenstein algebras
whose initial relation is of degree at most 3. Our result shows that those
Hilbert functions are exactly the so-called {\em SI-sequences} starting with
(1,4,h_2,h_3,...), where $h_4 \leq 33$. In particular, these Hilbert functions
are all unimodal.
  We also establish a more general unimodality result, which relies on the
values of the Hilbert function not being too big, but is independent of the
initial degree.
