Skip to main content

Computing Classical Modular Forms

  • Conference paper
  • First Online:
Arithmetic Geometry, Number Theory, and Computation

Abstract

We discuss practical and some theoretical aspects of computing a database of classical modular forms in the L-functions and modular forms database (LMFDB).

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 189.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 249.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 249.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. A. O. L. Atkin and Joseph Lehner, Hecke operators on Γ0(m), Math. Ann. 185 (1970), 134–160.

  2. A. O. L. Atkin and Wen-Ch’ing Winnie Li, Twists of newforms and pseudo-eigenvalues of W-operators, Invent. Math. 48 (1978), no. 3, 221–243.

  3. B. Banwait and J. Cremona, Tetrahedral elliptic curves and the local-to-global principle for isogenies, Algebra & Number Theory 8 (2014), no. 5, 1201–1229.

    Article  MathSciNet  Google Scholar 

  4. Karim Belabas and Henri Cohen, Modular forms in Pari/GP, Res. Math. Sci. 5 (2018), no. 3, Paper No. 37, 19 pp.

  5. Manjul Bhargava and Eknath Ghate, On the average number of octahedral newforms of prime level, Math. Ann. 344 (2009), no. 4, 749–768.

    Article  MathSciNet  Google Scholar 

  6. B. J. Birch, Elliptic curves over ℚ: A progress report, 1969 Number Theory Institute (State Univ. New York, Stony Brook, N.Y., 1969), Proc. Sympos. Pure Math., vol. 20, Amer. Math. Soc., Providence, 1971, 396–400.

  7. B. J. Birch, Hecke actions on classes of ternary quadratic forms, Computational number theory (Debrecen, 1989), de Gruyter, Berlin, 1991, 191–212.

    MATH  Google Scholar 

  8. Jonathan Bober, mflib software library, available at https://github.com/jwbober/mflib, 2019.

  9. Andrew R. Booker, Min Lee, and Andreas Strömbergsson, Twist-minimal trace formulas and the Selberg eigenvalue conjecture, J. Lond. Math. Soc. 102 (2020), no. 3, 1067–1134.

    Article  MathSciNet  Google Scholar 

  10. Andrew R. Booker, Artin’s conjecture, Turing’s method, and the Riemann hypothesis, Exp. Math. 15 (2006), no. 4, 385–408.

    Article  Google Scholar 

  11. Wieb Bosma, John Cannon, and Catherine Playoust, The Magma algebra system. I. The user language, J. Symbolic Comput. 24 (1997) no. 3–4, 235–265.

  12. Peter Bruin, Computing coeflcients of modular forms, Actes de la Conférence “Théorie des Nombres et Applications”, 19–36, Publ. Math. Besançon Algèbre Théorie Nr., 2011, Presses Univ. Franche-Comté, Besançon, 2011.

  13. Joe Buhler, An icosahedral modular form of weight one, Modular functions of one variable V, eds. Jean-Pierre Serre and Don Bernard Zagier, Lecture Notes in Math., vol. 601, Springer, Berlin-Heidelberg, 1977, 289–294.

  14. Joe P. Buhler, Icosahedral Galois representations, Lecture Notes in Math., vol. 654, Springer-Verlag, Berlin-New York, 1978.

  15. Joe P. Buhler, Benedict H. Gross, and Don B. Zagier, On the conjecture of Birch and Swinnerton-Dyer for an elliptic curve of rank 3, Math. Comp. 44 (1985), 473–481.

    Article  MathSciNet  Google Scholar 

  16. Jan Büthe, A method for proving the completeness of a list of zeros of certain L-functions, Math. Comp. 84 (2015), no. 295, 2413–2431.

    Article  MathSciNet  Google Scholar 

  17. Kevin Buzzard, Dimension of spaces of Eisenstein series, preprint available at http://wwwf.imperial.ac.uk/~buzzard/maths/research/notes/dimension_of_spaces_of_eisenstein_series.pdf, 2012.

  18. Kevin Buzzard, Computing weight one modular forms over \(\mathbb {C}\) and \(\overline {\mathbb {F}}_p\), Computations with modular forms, Contrib. Math. Comput. Sci., vol. 6, Springer, Cham, 2014, 129–146.

    Google Scholar 

  19. Kevin Buzzard and Alan Lauder, A computation of modular forms of weight one and small level, Ann. Math. Qué. 41 (2017), no. 2, 213–219.

    Article  MathSciNet  Google Scholar 

  20. Kevin Buzzard and William A. Stein, A mod five approach to modularity of icosahedral Galois representations, Pacific J. Math. 203 (2002), no. 2, 265–282.

    Article  MathSciNet  Google Scholar 

  21. Sarvadaman Chowla, The Riemann hypothesis and Hilbert’s tenth problem, Mathematics and Its Applications, vol. 4, Gordon and Breach, New York, 1965.

    MATH  Google Scholar 

  22. Henri Cohen, Francisco Diaz y Diaz, and Michel Olivier, Construction of tables of quartic fields, Algorithmic number theory (Leiden, 2000), Lecture Notes in Comput. Sci., vol. 1838, Springer, Berlin, 2000, 257-268.

  23. Henri Cohen, Francisco Diaz y Diaz, and Michel Olivier, Constructing complete tables of quartic fields using Kummer theory, Math. Comp. 72 (2003), no. 242, 941–951.

    Article  MathSciNet  Google Scholar 

  24. H. Cohen and J. Oesterlé, Dimensions des espaces de formes modulaires, Modular functions of one variable VI, eds. Jean-Pierre Serre and Don Zagier, Lecture Notes in Math., vol. 627, Springer, Berlin, 1977, 69–78.

  25. Henri Cohen and Fredrik Strömberg, Modular forms: a classical approach, Grad. Studies in Math., vol. 179, Amer. Math. Soc., Providence, 2017.

  26. Edgar Costa, Nicolas Mascot, Jeroen Sijsling, and John Voight, Rigorous computation of the endomorphism ring of a Jacobian, Math. Comp. 88 (2019), 1303–1339.

    Article  MathSciNet  Google Scholar 

  27. Edgar Costa and David Platt, A generic L-function calculator for motivic L-functions, available at https://github.com/edgarcosta/lfunctions, 2019.

  28. John E. Cremona, Aurel Page, and Andrew V. Sutherland, Sorting and labelling integral ideals in a number field, https://arxiv.org/abs/2005.09491.

  29. John E. Cremona, Abelian varieties with extra twist, cusp forms, and elliptic curves over imaginary quadratic fields, J. London Math. Soc. (2) 45 (1992), no. 3, 404–416.

  30. John E. Cremona, Algorithms for modular elliptic curves, 2nd ed., Cambridge University Press, Cambridge, 1997.

    Google Scholar 

  31. John E. Cremona, The elliptic curve database for conductors to 130000, Algorithmic number theory, eds. Florian Hess, Sebastian Pauli, and Michael Pohst, Lecture Notes in Comput. Sci., vol. 4076, Springer, Berlin, 2006, 11–29.

  32. John E. Cremona, The L-functions and modular forms database project, Found. Comput. Math. 16 (2016), no. 6, 1541–1553.

    Article  MathSciNet  Google Scholar 

  33. Henri Darmon, Alan Lauder, and Victor Rotger, Stark points and p-adic iterated integrals attached to modular forms of weight one, Forum Math. Pi 3 (2015), e8, 95 pp.

    Article  MathSciNet  Google Scholar 

  34. Henri Darmon, Alan Lauder, and Victor Rotger, First order p-adic deformations of weight one newforms, L-functions and automorphic forms, Contrib. Math. Comput. Sci., vol. 10, Springer, Cham, 2017, 39–80.

  35. Henri Darmon, Alan Lauder, and Victor Rotger, Overconvergent generalised eigenforms of weight one and class fields of real quadratic fields, Adv. Math. 283 (2015), 130–142.

    Article  MathSciNet  Google Scholar 

  36. Pierre Deligne and Jean-Pierre Serre, Formes modulaires de poids 1, Ann. Sci. Éc. Norm. Sup. (4) 7 (1974), no. 4, 507–530.

  37. P. Deligne, Valeurs de fonctions L et périodes d’intégrales, appendix by N. Koblitz and A. Ogus, Automorphic forms, representations and L-functions (Oregon State Univ., Corvallis, Ore., 1977), Part 2, eds. A. Borel, W. Casselman, Proc. Sympos. Pure Math., vol. 33, Amer. Math. Soc., Providence, 1979, 313–346.

  38. Fred Diamond and Jerry Shurman, A first course in modular forms, Grad. Texts in Math., vol. 228, Springer-Verlag, New York, 2005.

  39. Bas Edixhoven and Jean-Marc Couveignes, Computational aspects of modular forms and Galois representations: how one can compute in polynomial time the value of Ramanujan’s tau at a prime, Annals of Math. Studies, vol. 176, Princeton University Press, Princeton, NJ, 2011.

  40. M. Eichler, Einige Anwendungen der Spurformel im Bereich der Modularkorrespondenzen, Math. Ann. 168 (1967), 128–137.

    Article  MathSciNet  Google Scholar 

  41. M. Eichler, The basis problem for modular forms and the traces of the Hecke operators, Modular functions of one variable I (Proc. Internat. Summer School, Univ. Antwerp, Antwerp, 1972), ed. W. Kuijk, Lecture Notes in Math., vol. 320, Springer, Berlin, 1973, 75–151.

  42. Stephan Ehlen and Fredrik Strömberg, modforms-db software package, available at https://github.com/sehlen/modforms-db/tree/refactor, 2014.

  43. Andreas Enge and Andrew V. Sutherland, Class invariants by the CRT method, Algorithmic number theory, Lecture Notes in Comput. Sci., vol. 6197, Springer, Berlin, 2010, 142–156.

  44. Daniel Fiorilli, On the non-vanishing of Dirichlet L-functions at the central point, Q. J. Math. 66 (2015), no. 2, 517–528.

    Article  MathSciNet  Google Scholar 

  45. B.H. Gross and D.B. Zagier, Heegner points and derivatives of L-series, Invent. Math. 84 (1986), 225–320.

    Article  MathSciNet  Google Scholar 

  46. Erich Hecke, Mathematische Werke, Göttingen, Vandenhoeck & Ruprecht, 1959.

    MATH  Google Scholar 

  47. Hiroaki Hijikata, Explicit formula of the traces of Hecke operators for Γ0(N), J. Math. Soc. Japan 26 (1974), 56–82.

    Article  MathSciNet  Google Scholar 

  48. Hiroaki Hijikata, Arnold K. Pizer, and Thomas R. Shemanske, The basis problem for modular forms on Γ0(N), Mem. Amer. Math. Soc. 82 (1989), no. 418.

  49. Henryk Iwaniec and Emmanuel Kowalski, Analytic number theory, Amer. Math. Soc. Colloquium Publications, vol. 53, Amer. Math. Soc., Providence, 2004.

  50. H. Iwaniec and P. Sarnak, Perspectives on the analytic theory of L-functions, GAFA 2000 (Tel Aviv, 1999), Geom. Funct. Anal. 2000, Special Volume, Part II, 705–741.

  51. Fredrik Johansson, Arb: a C library for ball arithmetic, ACM Communications in Computer Algebra 47 (2013), no. 4, 166–169.

    Google Scholar 

  52. John W. Jones and David P. Roberts, Timing analysis of targeted Hunter searches, Algorithmic number theory (Portland, OR, 1998), Lecture Notes in Comput. Sci., vol. 1423, Springer, Berlin, 1998, 412–423.

    Google Scholar 

  53. Kazuya Kato, p-adic Hodge theory and values of zeta functions of modular forms, Astérisque 295 (2004), 117–290.

  54. Lloyd J. P. Kilford, Modular forms: A classical and computational introduction, 2nd ed., Imperial College Press, London, 2015.

    MATH  Google Scholar 

  55. Ian Kiming and Xiang Dong Wang, Examples of 2-dimensional, odd Galois representations of A5-type over ℚ satisfying the Artin conjecture, On Artin’s conjecture for odd 2-dimensional representations, Lecture Notes in Math., vol. 1585, Springer, Berlin, 1994, 109–121.

  56. David Lowry-Duda, Visualizing modular forms, J. S. Balakrishnan et al. (eds.), Arithmetic geometry, number theory, and computation, Simons symposia, Springer, Cham, 2021, 539–560.

    Google Scholar 

  57. Chandrashekhar Khare and Jean-Pierre Wintenberger, Serre’s modularity conjecture (I), Invent. Math. 178 (2009), no. 3, 485–504.

    Article  MathSciNet  Google Scholar 

  58. Andrew Knightly and Charles Li, Traces of Hecke operators, Math. Surveys Monogr., vol. 133, Amer. Math. Soc., Providence, 2006.

  59. Wen-Ch’ing Winnie Li, Newforms and functional equations, Math. Ann. 212 (1975), no. 4, 285–315.

  60. The LMFDB Collaboration, The L-functions and Modular Forms Database, http://www.lmfdb.org, 2019.

  61. Ju. I. Manin, Parabolic points and zeta functions of modular curves, Izv. Akad. Nauk SSSR Ser. Mat., 36 (1972), 19–66, translated in Math.-USSR Izvestija, 6 (1972), no. 1, 19–64.

  62. Kimball Martin, The basis problem revisited, arXiv:1804.04234v2, 2019.

  63. Nicolas Mascot, Certification of modular Galois representations, Math. Comp. 87 (2018), 381–423.

    Article  MathSciNet  Google Scholar 

  64. Loïc Merel, Universal Fourier expansions of modular forms, On Artin’s conjecture for odd 2-dimensional representations, Lecture Notes in Math., vol. 1585, Springer, Berlin, 1994, 59–94.

  65. J.-F. Mestre, La méthode des graphes. Exemples et applications, Proceedings of the international conference on class numbers and fundamental units of algebraic number fields (Katata, 1986), Nagoya Univ., Nagoya, 1986, 217–242.

    Google Scholar 

  66. Christian Meyer, Newforms of weight two for Γ0(N) with rational coefficients, http://meyer-idstein.de/weight2.pdf, 2005.

  67. Toshitsune Miyake, Modular forms, Springer Monographs in Math., Springer-Verlag, Berlin, 2006.

  68. Fumiyuki Momose, On the l-adic representations attached to modular forms, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 1, 89–109.

    MathSciNet  MATH  Google Scholar 

  69. Takeshi Ogasawara, Octahedral newforms of weight one associated to three-division points of elliptic curves, Funct. Approx. Comment. Math. 49 (2013), no. 1, 103–109.

    Article  MathSciNet  Google Scholar 

  70. Sami Omar, Non-vanishing of Dirichlet L-functions at the central point, Algorithmic Number Theory (ANTS 2008), Lecture Notes in Comp. Sci. 5011 (2008) 443–453.

    Article  MathSciNet  Google Scholar 

  71. Modular functions of one variable IV, Proceedings of the International Summer School on Modular Functions of One Variable and Arithmetical Applications, RUCA, University of Antwerp, Antwerp, July 17–August 3, 1972, eds. Bryan J. Birch and Willem Kuyk, Lecture Notes in Math., vol. 476, Springer-Verlag, Berlin, 1975.

  72. The PARI-group, Pari/GP (versions 2.11 and 2.12), Univ. Bordeaux, available at http://pari.math.u-bordeaux.fr, 2019.

  73. Arnold Pizer, An algorithm for computing modular forms on Γ0(N), J. Algebra 64 (1980), no. 2, 340–390.

    Article  MathSciNet  Google Scholar 

  74. Alexandru Popa, On the trace formula for Hecke operators on congruence subgroups, II, Res. Math. Sci. 5 (2018), no. 1, Paper No. 3, 24 pp.

  75. Kenneth A. Ribet, Galois representations attached to eigenforms with Nebentypus, Modular functions of one variable V, eds. Jean-Pierre Serre and Don Bernard Zagier, Lecture Notes in Math., vol. 601, Springer, Berlin-Heidelberg, 1977, 17–51.

  76. Kenneth A. Ribet, Twists of modular forms and endomorphisms of abelian varieties, Math. Ann. 253 (1980), no. 1, 43–62.

    Article  MathSciNet  Google Scholar 

  77. Kenneth A. Ribet, Mod p Hecke operators and congruences between modular forms, Invent. Math. 71 (1983), no. 1, 193–205.

    Article  MathSciNet  Google Scholar 

  78. Kenneth A. Ribet, Abelian varieties over ℚ and modular forms, Modular curves and abelian varieties, eds. John E. Cremona, Joan-C. Lario, Jordi Quer, and Kenneth A. Ribet, Progress in Math. 224, Birkhäuser, Basel, 2004, 241–261.

  79. The Sage Developers, SageMath (version 8.8), available at https://www.sagemath.org, 2019.

  80. George J. Schaeffer, Hecke stability and weight 1 modular forms, Math. Z. 281 (2015), no. 1–2, 159–191.

    MathSciNet  MATH  Google Scholar 

  81. A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. (N.S.) 20 (1956), 47–87.

  82. Jean-Pierre Serre, A course in arithmetic, Grad. Texts in Math. 7, Springer-Verlag, New York, 1973.

  83. J. P. Serre, Modular forms of weight one and Galois representations, Algebraic number fields: L-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975), ed. A. Fröhlich, Academic Press, London, 1977, 193–268.

  84. Jean-Pierre Serre, On a theorem of Jordan, Bull. Amer. Math. Soc. 40 (2003), no. 4, 429–440.

    Article  MathSciNet  Google Scholar 

  85. Rene Schoof and Marcel van der Vlugt, Hecke operators and the weight distribution of certain codes, J. Combin. Ser. A. 57 (1991) no. 2, 163–186.

    Article  MathSciNet  Google Scholar 

  86. Goro Shimura, Introduction to the arithmetic theory of automorphic functions, Kano Memorial Lectures, No. 1, Publications of the Mathematical Society of Japan, no. 1, Princeton Univ. Press, Princeton, N.J., 1971.

  87. Nils-Peter Skoruppa and Don Zagier, Jacobi forms and a certain space of modular forms, Invent. Math. 94 (1988), no. 1, 113–146.

    Article  MathSciNet  Google Scholar 

  88. William Stein, The Modular Forms Database, available at http://wstein.org/Tables/.

  89. William A. Stein, Modular forms, a computational approach, with an appendix by Paul E. Gunnells, Graduate Studies in Math., vol. 79, Amer. Math. Soc., Providence, 2007.

  90. William A. Stein, An introduction to computing modular forms using modular symbols, Algorithmic number theory: lattices, number fields, curves and cryptography, Math. Sci. Res. Inst. Publ., vol. 44, Cambridge Univ. Press, Cambridge, 2008, 641–652.

  91. Jacob Sturm, On the congruence of modular forms, Number theory (New York, 1984-1985), Lecture Notes in Math., vol. 1240, Springer, Berlin, 1987, 275–280.

  92. Andrew V. Sutherland, Computing Hilbert class polynomials with the Chinese remainder theorem, Math. Comp. 80 (2011), no. 273, 501–538.

    MathSciNet  MATH  Google Scholar 

  93. Andrew V. Sutherland, Accelerating the CM method, LMS J. Comput. Math. 15 (2012), 172–204.

    MathSciNet  Google Scholar 

  94. John Tate, Collected works of John Tate, Part I (1951-1975), eds. Barry Mazur and Jean-Pierre Serre, Amer. Math. Soc., Providence, 2016.

  95. Dave J. Tingley, Elliptic curves uniformized by modular functions, Ph.D. thesis, University of Oxford, 1975.

    Google Scholar 

  96. John Voight and David Zureick-Brown, The canonical ring of a stacky curve, accepted to Mem. Amer. Math. Soc., preprint available at https://arxiv.org/abs/1501.04657v3, 2019.

  97. Hideo Wada, Tables of Hecke operators. I, Seminar on Modern Methods in Number Theory (Inst. Statist. Math., Tokyo, 1971), Paper No. 39, Inst. Statist. Math., Tokyo, 1971, 1–10.

    Google Scholar 

  98. Hideo Wada, A table of Hecke operators. II, Proc. Japan Acad. 49 (1973), no. 6, 380–384.

  99. Mark Watkins, A discursus on 21 as a bound for ranks of elliptic curves over ℚ, and sundry related topics, available at https://magma.maths.usyd.edu.au/~watkins/papers/DISCURSUS.pdf, 2015.

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Andrew V. Sutherland .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Best, A.J. et al. (2021). Computing Classical Modular Forms. In: Balakrishnan, J.S., Elkies, N., Hassett, B., Poonen, B., Sutherland, A.V., Voight, J. (eds) Arithmetic Geometry, Number Theory, and Computation. Simons Symposia. Springer, Cham. https://doi.org/10.1007/978-3-030-80914-0_4

Download citation

Publish with us

Policies and ethics