  We show an intrinsic version of Thomason's fixed point theorem. Then we
determine the local structure of the Hilbert scheme of $\leq 7$ points in
$\mathbb{A}^3$. In particular, we show that in these cases the points with the
same extra dimension have the same singularity type. Using these results we
compute the equivariant Hilbert functions at the singularities and verify a
conjecture of Zhou on the Euler characteristics of tautological sheaves on
Hilbert schemes of points on $\mathbb{P}^3$ for $\leq 6$ points.
