Monatshefte für Mathematik

, Volume 188, Issue 1, pp 121–130 | Cite as

Computing bases of modular forms using the graded algebra structure

  • Michael O. Lam
  • Noah S. McClelland
  • Matthew R. Petty
  • John J. B. WebbEmail author


We develop a new algorithm to compute a basis for \(M_k(\Gamma _0(N))\), the space of weight k holomorphic modular forms on \(\Gamma _0(N)\), in the case when the graded algebra of modular forms over \(\Gamma _0(N)\) is generated at weight two. Our tests show that this algorithm significantly outperforms a commonly used algorithm which relies more heavily on modular symbols.


Holomorphic modular forms on \(\Gamma _0(N)\) Bases of modular forms Eta-quotients 

Mathematics Subject Classification

Primary 11F11 11F30 11Y16 Secondary 11F20 



The second and third authors received funding for this project through the James Madison University Tickle Scholarship Fund, while the fourth author received support from a James Madison University College of Science and Mathematics Faculty Summer Grant.


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Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Computer ScienceJames Madison UniversityHarrisonburgUSA
  2. 2.Department of Mathematics and StatisticsJames Madison UniversityHarrisonburgUSA

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