  We prove asymptotic formulas for the complex coefficients of $(\zeta
q;q)_\infty^{-1}$, where $\zeta$ is a root of unity, and apply our results to
determine secondary terms in the asymptotics for $p(a,b,n)$, the number of
integer partitions of $n$ with largest part congruent $a$ modulo $b$. Our
results imply that, as $n \to \infty$, the difference $p(a_1,b,n)-p(a_2,b,n)$
for $a_1 \neq a_2$ oscillates like a cosine, when renormalized by elementary
functions. Moreover, we give asymptotic formulas for arbitrary linear
combinations of $\{p(a,b,n)\}_{1 \leq a \leq b}$.
