  The Quantum Modularity Conjecture of Zagier predicts the existence of a
formal power series with arithmetically interesting coefficients that appears
in the asymptotics of the Kashaev invariant at each root of unity. Our goal is
to construct a power series from a Neumann-Zagier datum (i.e., an ideal
triangulation of the knot complement and a geometric solution to the gluing
equations) and a complex root of unity $\zeta$. We prove that the coefficients
of our series lie in the trace field of the knot, adjoined a complex root of
unity. We conjecture that our series are those that appear in the Quantum
Modularity Conjecture and confirm that they match the numerical asymptotics of
the Kashaev invariant (at various roots of unity) computed by Zagier and the
first author. Our construction is motivated by the analysis of singular limits
in Chern-Simons theory with gauge group $SL(2,C)$ at fixed level $k$, where
$\zeta^k=1$.
