Abstract
Given a complex number a, we prove that there is at most one normalized Hecke eigenform of level one whose third Fourier coefficient is given by a, assuming that the characteristic polynomials of the Hecke operators \(T_3\) are irreducible. This generalizes a result of Vilardi and Xue on the second Fourier coefficients. Under the assumption on irreducibility of characteristic polynomials of \(T_2\) and \(T_3\), we further show that there is at most one normalized eigenform of level one whose second or third Fourier coefficient is given by a.
Similar content being viewed by others
References
Conrey, J.B., Farmer, D.W., Wallace, P.J.: Factoring Hecke polynomials modulo a prime. Pacific J. Math. 196(1), 123–130 (2000)
Dong, P.: Minoration de combinaisons linéaires de deux logarithmes \(p\)-adiques. Ann. Fac. Sci. Toulouse Math. (5) 12(2), 195–250 (1991)
Ghitza, A.: Distinguishing Hecke eigenforms. Int. J. Number Theory 7(5), 1247–1253 (2011)
Ghitza, A., McAndrew, A.: Experimental evidence for Maeda’s conjecture on modular forms. Tbil. Math. J. 5(2), 55–69 (2012)
Hatada, K.: Eigenvalues of Hecke operators on \({\rm SL}(2,\,{ Z})\). Math. Ann. 239(1), 75–96 (1979)
Hida, H., Maeda, Y.: Non-abelian base change for totally real fields. In: Pacific J. Math. 1997, pp. 189–217
Knightly, A., Li, C.: Traces of hecke operators. In: Mathematical Surveys and Monographs, vol. 133. American Mathematical Society, Providence, RI, (2006)
Murty, M.R., Congruences between modular forms. In Analytic number theory (Kyoto, : vol. 247 of London Math. Soc. Lecture Note Ser. Cambridge Univ. Press, Cambridge 1997, 309–320 (1996)
Ram Murty, M., Kumar Murty, V.: Odd values of Fourier coefficients of certain modular forms. Int. J. Number Theory 3(3), 455–470 (2007)
Rouse, J.: Vanishing and non-vanishing of traces of Hecke operators. Trans. Amer. Math. Soc. 358(10), 4637–4651 (2006)
Sturm, J.: On the congruence of modular forms. In: Number theory (New York, 1984–1985), vol. 1240 of Lecture Notes in Math. Springer, Berlin, 1987, pp. 275–280
Vilardi, T., Xue, H.: Distinguishing eigenforms of level one. Int. J. Number Theory 14(1), 31–36 (2018)
Yu, K.R.: Linear forms in \(p\)-adic logarithms. Acta Arith. 53(2), 107–186 (1989)
Yu, K.R.: Linear forms in \(p\)-adic logarithms. II. Compositio Math. 74(1), 15–113 (1990)
Zagier, D.: Traces des opérateurs de Hecke. In: Séminaire Delange-Pisot-Poitou, 17e année: 1975/76. Théorie des nombres: Fasc. 2, Exp. No. 23. 1977, p. 12
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Xue, H., Zhu, D. Uniqueness of Fourier coefficients of eigenforms. Ramanujan J 57, 487–505 (2022). https://doi.org/10.1007/s11139-021-00450-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11139-021-00450-7