  Andrews, Dyson and Hickerson proved many interesting properties of
coefficients for a Ramanujan's $q$-hypergeometric series by relating it to real
quadratic field $\Q(\sqrt{6})$ and using the arithmetic of $\Q(\sqrt{6})$,
hence solved a conjecture of Andrews on the distributions of its Fourier
coefficients. Motivated by Andrews's conjecture, we discuss an interesting
$q$-hypergeometric series which comes from a Lerch sum and rank and crank
moments for partitions and overpartitions. We give Andrews-like conjectures for
its coefficients. We obtain partial results on the distributions of small
values of its coefficients toward these conjectures.
