To expand on wccanard's comment : see my answer to [this question][1] for a general method to compute a rational function with given (principal) divisor.

This will give you $f_e$ as a rational function of $x$ and $y$. You then just need to expand this in terms of the standard formal coordinate $z=-x/y$.


  [1]: https://mathoverflow.net/questions/121291/calculate-function-from-its-divizor/