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Computing bases of modular forms using the graded algebra structure

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Abstract

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.

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Acknowledgements

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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Authors and Affiliations

  1. Department of Computer Science, James Madison University, MSC 4103, Harrisonburg, VA, 22807, USA

    Michael O. Lam

  2. Department of Mathematics and Statistics, James Madison University, MSC 1911, Harrisonburg, VA, 22807, USA

    Noah S. McClelland, Matthew R. Petty & John J. B. Webb

Authors
  1. Michael O. Lam
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  2. Noah S. McClelland
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  3. Matthew R. Petty
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  4. John J. B. Webb
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Corresponding author

Correspondence to John J. B. Webb.

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Communicated by A. Constantin.

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Lam, M.O., McClelland, N.S., Petty, M.R. et al. Computing bases of modular forms using the graded algebra structure. Monatsh Math 188, 121–130 (2019). https://doi.org/10.1007/s00605-018-1168-9

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  • DOI: https://doi.org/10.1007/s00605-018-1168-9

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