Functiones et Approximatio Commentarii Mathematici

Fourier coefficients of Hecke eigenforms

Ronald Evans

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We provide systematic evaluations, in terms of binary quadratic representations of $4p$, for the $p$-th Fourier coefficients of each member $f$ of an infinite class $\mathcal{C}$ of CM eigenforms. As an application, previously conjectured evaluations of three algebro-geometric character sums can now be formulated explicitly without reference to eigenforms. There are several non-CM newforms that appear to share some properties with the eigenforms in $\mathcal{C}$, and we pose some conjectures about their Fourier coefficients.

Article information

Funct. Approx. Comment. Math., Volume 46, Number 2 (2012), 147-159.

First available in Project Euclid: 25 June 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11F11: Holomorphic modular forms of integral weight
Secondary: 11F30: Fourier coefficients of automorphic forms 11E25: Sums of squares and representations by other particular quadratic forms 11R29: Class numbers, class groups, discriminants

Hecke eigenforms Hecke characters newforms nebentypus genus class group ideal class group imaginary quadratic field Kloosterman sums representation of primes by binary quadratic forms


Evans, Ronald. Fourier coefficients of Hecke eigenforms. Funct. Approx. Comment. Math. 46 (2012), no. 2, 147--159. doi:10.7169/facm/2012.46.2.1.

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