  Let $\mathbb{K} = \mathbb{Q}(\sqrt{-d})$ be an imaginary quadratic number
field of class number $1$ and $\mathcal{O}_{\mathbb{K}}$ its ring of integers.
We study a family of Hecke $L$-functions associated to angular characters on
the non-zero ideals of $\mathcal{O}_{\mathbb{K}}$. Using the powerful Ratios
Conjecture (RC) due to Conrey, Farmer, and Zirnbauer, we compute a conditional
asymptotic for the average $1$-level density of the zeros of this family,
including terms of lower order than the main term in the Katz-Sarnak Density
Conjecture coming from random matrix theory. We also prove an unconditional
result about the $1$-level density, which agrees with the RC prediction when
our test functions have Fourier transforms with support in $(-1,1)$.
