I am looking for an algorithm to compute the period polynomial $$P(z,f) := \int_C f(\tau) (z-\tau)^{k-2} d \tau$$ for a cusp form $f(\tau)$ of weight-k, where $C$ is a path connecting $\tau =0$ and $\tau = i \infty$. I am interested in a case where $f(\tau)$ is meromorphic, and is allowed to have a pole in the upper half plane (e.g., $f = (E_4^3-E_6^2)/E_6$ and $k=6$).
Although the coefficients appearing in $P(z,f)$ are known to be the special values of the Mellin transform of $f$, I am looking for an algorithm to compute those coefficients and express them in terms of rational numbers and $\zeta(n)$'s, if possible. I saw references that seem to do this task in the cases $f$ is an Eisenstein series, or a theta function modified by characters, but I have no idea how to do this task for more general $f$'s (such as the one above). Any help or suggestion is appreciated.