Extending support for the centered moments of the low lying zeroes of cuspidal newforms

We study low-lying zeroes of \(L\)-functions and their \(n\)-level density, which relies on a smooth test function \(\phi\) whose Fourier transform \(\widehat\phi\) has compact support. Assuming the generalized Riemann hypothesis, we compute the \(n^\text{th}\) centered moments of the \(1\)-level density of low-lying zeroes of \(L\)-functions associated with weight \(k\), prime level \(N\) cuspidal newforms as \(N \to \infty\), where \({\rm supp}(\widehat\phi) \subset \left(-2/n, 2/n\right)\). The Katz-Sarnak density conjecture predicts that the \(n\)-level density of certain families of \(L\)-functions is the same as the distribution of eigenvalues of corresponding families of orthogonal random matrices. We prove that the Katz-Sarnak density conjecture holds for the \(n^\text{th}\) centered moments of the 1-level density for test functions with \(\widehat{\phi}\) supported in \(\left(-2/n, 2/n\right)\), for families of cuspidal newforms split by the sign of their functional equations. Our work provides better bounds on the percent of forms vanishing to a certain order at the central point. Previous work handled the 1-level for support up to 2 and the \(n\)-level up to \(\min(2/n, 1/(n-1))\); we are able to remove the second restriction on the support and extend the result to what one would expect, based on the 1-level, by finding a tractable vantage to evaluate the combinatorial zoo of terms which emerge.