  In this paper, we discuss the refined Humbert invariant for abelian product
surfaces with complex multiplication. More precisely, we give the
classification of quadratic forms which are equivalent to some refined Humber
invariant $q_{(A,\theta)},$ where $A$ is isogenous to $E\times E,$ where $E$
has complex multiplication, and $q_{(A,\theta)}$ is an imprimitive form. To
give this classification, we use and derive some results in the theory of
integral ternary and binary quadratic forms.
