Lower Order Terms for the One-Level Density of a Symplectic Family of Hecke L-Functions

In this paper we apply the \(L\)-function Ratios Conjecture to compute the one-level density for a symplectic family of \(L\)-functions attached to Hecke characters of infinite order. When the support of the Fourier transform of the corresponding test function \(f\) reaches \(1\), we observe a transition in the main term, as well as in the lower order term. The transition in the lower order term is in line with behavior recently observed by D. Fiorilli, J. Parks, and A. S\"odergren in their study of a symplectic family of quadratic Dirichlet \(L\)-functions. We then directly calculate main and lower order terms for test functions \(f\) such that supp(\(\widehat{f}) \subset [-\alpha,\alpha]\) for some \(\alpha