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Building on the work of Iwaniec, Luo and Sarnak, we use the $n$-level densityto bound the probability of vanishing to order at least $r$ at the centralpoint for families of cuspidal newforms of prime level $N \to \infty$, split bysign. There are three methods to improve bounds on the order of vanishing:optimizing the test functions, increasing the support, and increasing the$n$-level density studied. Previous work determined the optimal test functionsfor the $1$ and $2$-level densities in certain support ranges, leading tomarginal improvements in bounds and making it not a productive avenue forfurther research. Similarly the support has been increased as far as possible,and further progress is shown to be related to delicate and difficultconjectures in number theory. Thus we concentrate on the third method, andstudy the higher centered moments (which are similar to the $n$-level densitiesbut combinatorially easier). We find the level at each rank for which thebounds on the order of vanishing is the best, thus producing world-recordbounds on the order of vanishing to rank at least $r$ for every $r > 2$ (forexample, our bounds for vanishing to order at least 5 or at least 6 are lessthan half the previous bounds, a significant improvement). Additionally, weexplicitly calculate the optimal test function for the $1$-level density fromprevious work and compare it to the naive test functions for higher levels. Indoing so, we find that the optimal test function for certain levels are not theoptimal for other levels, and some test functions may outperform others forsome levels but not in others. Finally, we explicitly calculate the integralsneeded to determine the bounds, doing so by transforming an $n$-dimensionalintegral to a $1$-dimensional integral and greatly reducing the computationcost in the process.

Bounding the Order of Vanishing of Cuspidal Newforms via the nth Centered Moments