  We study the harmonically weighted one-level density of low-lying zeros of
$L$-functions in the family of holomorpic newforms of fixed even weight $k$ and
prime level $N$ tending to infinity. For this family, Iwaniec, Luo and Sarnak
proved that the Katz--Sarnak prediction for the one-level density holds
unconditionally when the support of the Fourier transform of the implied test
function is contained in $(-\tfrac32,\tfrac32)$. In this paper, we extend this
admissible support to $(-\Theta_k,\Theta_k)$, where $\Theta_2 = 1.866\dots$ and
$\Theta_k$ tends monotonically to $2$ as $k$ tends to infinity. This is
asymptotically as good as the best known GRH result. The main novelty in our
analysis is the use of zero-density estimates for Dirichlet $L$-functions.
