QEta 4.0

Abstract

QEta is a software package that implements algorithms in connection with series expansions of modular functions at cusps with respect to the groups \(\mathrm{SL}_2(\mathbb{Z})\), \(\Gamma_0(N)\), \(\Gamma_1(N)\), and \(\Gamma(N)\).

With QEta you can

QEta is programmed in the computer algebra system FriCAS.

Citation

If you use QEta in your work, please cite it as:

@Misc{QEta,
   key =          {QEta 4.0},
   author =       {Ralf Hemmecke},
   year =         {2025},
   title =        {{Q}{E}ta---{A} {FriCAS} package to compute with
                   $q$-expansions of modular functions},
   note =         {Available at \url{https://risc.jku.at/sw/qeta} and
                   \url{https://hemmecke.github.io/qeta}}
 }

Overview

The QEta package started at the end of 2015 with an implementation of the AB algorithm from the article “An algorithmic approach to Ramanujan-Kolberg identities” by Silviu Radu (2015) and the Samba algorithm from the article “Dancing Samba with Ramanujan Partition Congruences” by Ralf Hemmecke (2018).

In addition QEta implements the algorithm from the article “Construction of all Polynomial Relations among Dedekind Eta Functions of Level N” by Hemmecke and Radu to compute all polynomial relations of Dedekind eta-functions of a certain level (related website: Ideal of Relations of eta Functions).

The computations for the article “Construction of Modular Function Bases for Γ₀(121) related to p(11n+6)” by Hemmecke, Paule, and Radu were done with the QEta package. See also the related website.

In 2021 the extensions to Radu’s algorithm for generalized eta-quotients described in the the article “Finding Modular Functions for Ramanujan-Type Identities” by Chen, Du, and Zhao have also been included in QEta.

In 2023, the implementation of an extension of the Samba algorithm to the multidimensional case started. The resulting algortithm, MultiSamba, can be used to compute a module basis \(b_0,\ldots,b_s\) of \(t, f_1,\ldots,f_r \in R^n\) (for \(n\ge1\)) such that

\[C[t, f_1,...,f_r] = \langle b_0, \ldots, b_s\rangle_{C[t]}.\]

where \(R=C[q]\) or \(R=C((q))\) (in the latter case with some restrictions to ensure termination). It can find the modular equation of two modular functions and has been employed for the automatic algebraic derivation of Ramanujan-Sato series, i.e., formulas such as the well known formula

\[\begin{align*} \frac{1}{\pi} &= \frac{2\sqrt{2}}{99^2} \sum_{n=0}^\infty \frac{(4n)!}{n!^4} \frac{26390n+1103}{396^{4n}}. \end{align*}\]

and many more of this kind, see RISC Report 16-06 and this website for more details.

The underlying theory of the programs is described in the above articles which are also available as RISC reports 15-14, 16-06, 18-03. 19-10, 25-01, and this preprint. Among other result 16-06 contains a witness relation in terms of Dedekind eta functions for the Ramanujan congruence

\[p(11 n + 6) \equiv 0 \pmod{11}\]

for the partition function \(p(n)\) that is given through its generating series.

\[\sum_{n=0}^\infty p(n) q^n = \prod_{n=1}^\infty (1 − q^n)^{-1}\]

More deails and a description of how to work with QEta can be found in the QEtaTutorial and other Tutorials

The QEta package requires FriCAS 1.3.12. See FriCAS download and the FriCAS installation guide for a description of how to get and install FriCAS.

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