  In this paper, we prove that for any totally real field $F$, weight $k$, and
nebentypus character $\chi$, the proportion of Hilbert cusp forms over $F$ of
weight $k$ and character $\chi$ with bounded field of rationality approaches
zero as the level grows large. This answers, in the affirmative, a question of
Serre. The proof has three main inputs: first, a lower bound on fields of
rationality for admissible $GL_2$ representations; second, an explicit
computation of the (fixed-central-character) Plancherel measure for $GL_2$; and
third, a Plancherel equidsitribution theorem for cusp forms with fixed central
character. The equidistribution theorem is the key intermediate result and
builds on earlier work of Shin and Shin-Templier and mirrors work of
Finis-Lapid-Mueller by introducing an explicit bound for certain families of
orbital integrals.
