  In this paper, we evaluate explicitly certain quadratic Hecke Gauss sums of
$\mathbb{Q}(\omega), \omega=\exp \left( \frac {2\pi i}{3}\right)$. As
applications, we study the moments of central values of quadratic Hecke
$L$-functions of $\mathbb{Q}(\omega)$, and establish quantitative non-vanishing
result for the $L$-values. We also establish an one level density result for
the low-lying zeros of quadratic Hecke $L$-functions of $\mathbb{Q}(\omega)$.
