  We study the one-level density for families of L-functions associated with
cubic Dirichlet characters defined over the Eisenstein field. We show that the
family of $L$-functions associated with the cubic residue symbols $\chi_n$ with
$n$ square-free and congruent to 1 modulo 9 satisfies the Katz-Sarnak
conjecture for all test functions whose Fourier transforms are supported in
$(-13/11, 13/11)$, under GRH. This is the first result extending the support
outside the \emph{trivial range} $(-1, 1)$ for a family of cubic L-functions.
This implies that a positive density of the L-functions associated with these
characters do not vanish at the central point $s=1/2$. A key ingredient in our
proof is a bound on an average of generalized cubic Gauss sums at prime
arguments, whose proof is based on the work of Heath-Brown and Patterson.
