The Ramanujan Journal

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

The second Dirichlet coefficient starts out negative

  • David W. Farmer
  • Sally Koutsoliotas


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 


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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

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

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