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.
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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
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DOI: https://doi.org/10.1007/s11139-021-00450-7