From the proof of Weil's conjectures it follows that $|q^k - \# X(\mathbb{F}_{q^k})| = O(q^{k(n - \frac{1}{2})})$, where $X$ is a smooth variety over $\mathbb{F}_q$ and $n = \dim X$ (see for example http://www-math.mit.edu/~poonen/papers/Qpoints.pdf). How the constant in $O(\ldots)$ can be estimated?
Every estimation (even very rough) is interesting.