Take a hyperelliptic curve $C$ (nonsingular, algebraically closed field) of genus $g$. The theta characteristic(s) of the curve are defined to be the line bundles $\{\theta\in Pic^{g-1}(C)|\theta^{\otimes 2} = \omega_C\}$, where $\omega_C$ is the canonical sheaf. We can (after fixing some $\theta$) write $\theta=\mathscr{O}_C(\Theta)$ and $\omega_C=\mathscr{O}_C(K)$ for the canonical divisor $K$ and corresponding theta divisor $\Theta$. It is well-known that there are $2^{2g}$ theta characteristics for a genus $g$ hyperelliptic curve.
Since we are in the nonsingular case we can consider $\Theta,K$ as Weil divisors. In particular we must have $[2\Theta]=[K]$. Naively I hoped to simply formally divide the coefficients in $K$ by $2$ to find a $\Theta$, but although the degree of $K$ is even, the coefficients of it are all not even (for example the coefficients at the Weierstrass points are $1$).
Question: How does one calculate $\theta,\Theta$ in practice? Are there any examples of this for low-genus curves? Any references?