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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.

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If $f$ is a cusp form of even integral weight $k\geq 4$ (I avoid $k=2$ because then the period polynomial is constant), then the "special values of the Mellin transform of $f$" are in fact critical values of the modular $L$-function $L(s,f)$. In general, these will not be expressible "in terms of rational numbers and $\zeta(n)$'s." This is different from the case of Eisenstein series or theta functions. A modern paper by Jin, Ma, Ono, and Soundararajan that discusses the period polynomials for such cusp forms (and more generally for holomorphic cuspidal newforms of even integral weight $k\geq 4$, level $N\geq 1$, and trivial nebentypus) can be found here.

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  • $\begingroup$ Thank you very much. References helped me a lot. For non-experts, it is also important to know when there is virtually no hope, as time is not going to be wasted too much then. So, when it comes to the common transcendental multipliers (denoted by $\omega_\pm$ in some literatures) or special values of L-functions, we may compute them numerically, but there are not much else we can do about them even today.... is that right? $\endgroup$ Aug 11, 2021 at 11:29

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