Spins, governing fields and negative Pell

Let $K$ be a cyclic, totally real extension of $\mathbb{Q}$ of degree at least 3 and let $\sigma$ be a generator of Gal$(K/\mathbb{Q})$. We further assume that the totally positive units are exactly the squares of units. In this case, Friedlander, Iwaniec, Mazur and Rubin define the spin of an odd principal ideal to be spin$(\sigma, a) = (\alpha/\sigma(\alpha))_K$, where $\alpha$ is a totally positive generator of $a$ and $(*/*)_K$ is the quadratic residue symbol in $K$. They then proceed to prove equidistribution of spin$(\sigma, p)$ as $p$ varies over the odd principal prime ideals of $K$. We will show how to extend this to general fields and mixed moments of spin symbols. As an application we show the non-existence of governing fields for the 16-rank of $\mathbb{Q}(\sqrt{-p})$ and discuss some progress towards  Stevenhagen's conjecture on the negative Pell equation.