The Ramanujan Journal

, Volume 41, Issue 1–3, pp 335–343 | Cite as

The second Dirichlet coefficient starts out negative



Classical modular forms of small weight and low level are likely to have a negative second Fourier coefficient. Similarly, the labeling scheme for elliptic curves tends to give smaller labels to the higher-rank curves. These observations are easily made when browsing the L-functions and Modular Forms Database, available at An explanation lies in the L-functions associated to these objects.


Modular form Negative coefficient L-function Elliptic curve 

Mathematics Subject Classification

11F11 11M26 11G05 


  1. 1.
    Breuil, C., Conrad, B., Diamond, F., Taylor, R.: On the modularity of elliptic curves over Q: wild 3-adic exercises. J. Am. Math. Soc. 14(4), 843–939 (2001)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Choie, Y.J., Kohnen, W.: The first sign change of Fourier coefficients of cusp forms. Am. J. Math. 131(2), 517–543 (2009)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Conrey, J.B., Duke, W., Farmer, D.W.: The distribution of the eigenvalues of Hecke operators. Acta Arith. 78(4), 405–409 (1997)MathSciNetMATHGoogle Scholar
  4. 4.
    Cremona, J: The elliptic curve database for conductors to 130000. In: ANTS-VII Proceedings. Lecture Notes in Computer Science, vol. 4076, pp. 11–29 (2006)Google Scholar
  5. 5.
    Farmer, D., Koutsoliotas, S., Lemurell, S.: Varieties via their L-functions. Preprint, arXiv:1502.00850
  6. 6.
    Farmer, D., Koutsoliotas, S., Lemurell, S.: L-functions with rational integer coefficients I: degree 4 and weight 0. PreprintGoogle Scholar
  7. 7.
    Iwaniec, H.: Topics in classical automorphic forms. In: Graduate Studies in Mathematics, vol. 17. American Mathematical Society, Providence. pp. xii+259 (1997). ISBN: 0-8218-0777-3Google Scholar
  8. 8.
    Iwaniec, H., Kohnen, W., Sengupta, J.: The first negative Hecke eigenvalue. Int. J. Number Theory 3(3), 355–363 (2007)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Iwaniec, H., Kowalski, E.: Analytic number theory. In: American Mathematical Society Colloquium Publications, vol. 53. American Mathematical Society, Providence (2004)Google Scholar
  10. 10.
    Kohnen, W., Sengupta, J.: On the first sign change of Hecke eigenvalues of newforms. Math. Z. 254(1), 173–184 (2006)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Kowalski, E., Lau, Y.-K., Soundararajan, K., Wu, J.: On modular signs. Math. Proc. Camb. Philos. Soc. 149(3), 389–411 (2010)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    The LMFDB Collaboration: The L-functions and Modular Forms Database. (2013). Accessed 7 Oct 2014
  13. 13.
    Murty, M.R.: Oscillations of Fourier coefficients of modular forms. Math. Ann. 262(4), 431–446 (1983)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Rubinstein, M.O.: Elliptic curves of high rank and the Riemann zeta function on the one line. Exp. Math. 22(4), 465–480 (2013)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Serre, J.-P.: Rpartition asymptotique des valeurs propres de l’oprateur de Hecke Tp. J. Am. Math. Soc. 10(1), 75–102 (1995). 11F30 (11F25 11G20 11N37 11R45)CrossRefGoogle Scholar
  16. 16.
    Sarnak, P.: Statistical properties of eigenvalues of the Hecke operators. In: Analytic Number Theory and Diophantine Problems (Stillwater, OK, 1984), Progr. Math., vol. 70, pp. 321–331. Birkhauser, Boston (1987)Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.American Institute of MathematicsSan JoseUSA
  2. 2.Bucknell UniversityLewisburgUSA

Personalised recommendations