Moments of one-level densities in families of holomorphic cusp forms in the level aspect

We study the $n^{\rm th}$ centered moments of the $1$-level density for the low-lying zeros of $L$-functions attached to holomorphic cuspidal newforms of large prime level and fixed weight. Assuming the Generalized Riemann Hypotheses, we compute this statistic for any $n\ge 1$ and for all test functions whose Fourier transforms are supported in $\left(-2/n, \, 2/n\right)$. This is believed to be the natural limit of the current technology. Our work significantly extends beyond the trivial range $(-1/n, \, 1/n)$ and surpasses the previous record of $(-1/(n-1),\, 1/(n-1))$ whenever $n>2$. The Katz-Sarnak philosophy predicts that the aforementioned statistic can be modeled by the corresponding statistic for the eigenvalues of random orthogonal matrices. We prove that this is the case for test functions with Fourier support contained in $(-2/n,\, 2/n)$. The main technical innovation is a tractable vantage to evaluate the combinatorial zoo of terms, similar to the work of Conrey-Snaith and Mason-Snaith. As an application, our work provides better bounds on the order of vanishing at the central point for the $L$-functions in our family.