Cohomology of structure sheaf of abelian variety

Question

Let $X$ be an abelian variety over $\mathbb{C}$ of dimension $n$. Consider the structure sheaf $O_X$. It's Euler characteristic is zero, because $\chi(O_X)= (O_X^n)/n!$. And the self intersection of $O_X$ is 0. But is there some way to compute the $h^i(X,O_X)$? If the dimension $n=2$, then since $h^0=h^2=1$, we get $h^1=2$. What about for n>2? Is there some way to check this dimension?