# Breaking Universality in the Lower Order Terms in the 1-level and 2-level Density of Holomorphic Cusp Newforms

## TL;DR

This paper investigates the lower-order terms in the 1-level and 2-level density of holomorphic cusp newforms, revealing that these terms depend on the arithmetic of the family and break the expected universality predicted by the Katz-Sarnak conjecture.

## Contribution

It extends Miller's work by explicitly computing family-dependent lower-order correction terms in the densities of holomorphic cusp newforms up to a refined error bound.

## Key findings

- Lower-order terms depend on the arithmetic of the family.
- Different prime factorizations of the level lead to different lower-order terms.
- The results break the universality of the main behavior predicted by the Katz-Sarnak conjecture.

## Abstract

The Katz-Sarnak density conjecture states that, as the analytic conductor $R \to \infty$, the distribution of the normalized low-lying zeros (those near the central point $s = 1/2$) converges to the scaling limits of eigenvalues clustered near 1 of subgroups of $U(N)$. There is extensive evidence supporting this conjecture for many families, including the family of holomorphic cusp newforms. Interestingly, there are very few choices for the main term of the limiting behavior. In 2009, S. J. Miller computed lower-order terms for the 1-level density of families of elliptic curve $L$-functions and compared to cuspidal newforms of prime level; while the main terms agreed, the lower order terms depended on the arithmetic of the family. We extend his work by identifying family-dependent lower-order correction terms in the weighted 1-level and 2-level densities of holomorphic cusp newforms up to $O\left(1/\log^4 R\right)$ error, sharpening Miller's $O\left(1/\log^3 R\right)$ error. We consider cases where the level is prime or when the level is a product of two, not necessarily distinct, primes. We show that the rates at which the prime factors of the level tend to infinity lead to different lower-order terms, breaking the universality of the main behavior.