Abstract
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 http://www.LMFDB.org/. An explanation lies in the L-functions associated to these objects.
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Dedicated to the memory of Marvin Knopp
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Farmer, D.W., Koutsoliotas, S. The second Dirichlet coefficient starts out negative. Ramanujan J 41, 335–343 (2016). https://doi.org/10.1007/s11139-015-9716-6
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DOI: https://doi.org/10.1007/s11139-015-9716-6