For many families of Hecke operators, the second coefficient of the characteristic polynomial never repeats, and this non-repetition yields a conditional uniqueness theorem for Fourier coefficients of modular forms.
Nonvanishing of second coefficients of Hecke polynomials
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Non-repetition of second coefficients of Hecke polynomials
For many families of Hecke operators, the second coefficient of the characteristic polynomial never repeats, and this non-repetition yields a conditional uniqueness theorem for Fourier coefficients of modular forms.