  The Density Conjecture of Katz and Sarnak associates a classical compact
group to each reasonable family of $L$-functions. Under the assumption of the
Generalized Riemann Hypothesis, Rubinstein computed the $n$-level density of
low-lying zeros for the family of quadratic Dirichlet $L$-functions in the case
that the Fourier transform $\hat{f}(u)$ of any test function $f$ is supported
in the region $\sum^n_{j=1}u_j < 1$ and showed that the result agrees with the
Density Conjecture. In this paper, we improve Rubinstein's result on computing
the $n$-level density for the Fourier transform $\hat{f}(u)$ being supported in
the region $\sum^n_{j=1}u_j < 2$.
