  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
$\alpha <1$, and observe that this unconditional result is in agreement with
the prediction provided by the Ratios Conjecture. As the analytic conductor of
these L-functions grow twice as large (on a logarithmic scale) as the
cardinality of the family in question, this is the optimal support that can be
expected with current methods. Finally as a corollary we deduce that, under
GRH, at least 75$\%$ of these $L$-functions do not vanish at the central point.
