  In this paper, we completely classify $3$-dimensional complete self-shrinkers
with constant norm $S$ of the second fundamental form and constant $f_{3}$ in
Euclidean space $\mathbb R^{4}$, where $h_{ij}$ are components of the second
fundamental form, $S=\sum_{i,j}h^{2}_{ij}$ and
$f_{3}=\sum_{i,j,k}h_{ij}h_{jk}h_{ki}$.
