Extending the unconditional support in an Iwaniec-Luo-Sarnak family

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 $(-\frac{3}{2},\frac{3}{2})$. In this paper, we extend this admissible support to $(-{\mathrm{\Theta }}_k,{\mathrm{\Theta }}_k)$, where ${\mathrm{\Theta }}_2=1.866\dots$ and ${\mathrm{\Theta }}_k$ 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.