  We classify all possible $h$-vectors of graded artinian Gorenstein algebras
in socle degree 4 and codimension $\leq 17$, and in socle degree 5 and
codimension $\leq 25$. We obtain as a consequence that the least number of
variables allowing the existence of a nonunimodal Gorenstein $h$-vector is 13
for socle degree 4, and 17 for socle degree 5.
  In particular, the smallest nonunimodal Gorenstein $h$-vector is
$(1,13,12,13,1)$, which was constructed by Stanley in his 1978 seminal paper on
level algebras. This solves a long-standing open question in this area. All of
our results are characteristic free.
