Let $A$ be a finite abelian group. Let $q:A\times A\to \mathbb{C}^{\times}$ be a non-degenerate bicharacter (that is: for every $a\in A$ $q(a,-)$ and $q(-,a)$ are characters of $A$, which are trivial if and only if $a=1$).

What can we say about the sum $$\sum_{a\in A}q(a,a)$$ (where the sum is being taken in $\mathbb{C}$?)

Can this sum be expressed by any invariants of $q$ or of $A$?