Fourier Coefficients of Hecke Eigenforms
نویسنده
چکیده
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 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 C, and we pose some conjectures about their Fourier coefficients.
منابع مشابه
The Hardness of Computing an Eigenform Eric Bach and Denis Charles
The Fourier coefficients of modular forms encode very interesting arithmetic data. For example, divisor sums, partition numbers, trace of Frobenius of the reduction modulo primes of an elliptic curve over Q, and more generally, trace of Frobenius of many Galois representations of dimension 2 over finite fields (this being a conjecture of Serre) are all known to be, or conjectured to be, Fourier...
متن کاملFe b 20 16 SIGN CHANGES OF FOURIER COEFFICIENTS OF MODULAR FORMS OF HALF INTEGRAL WEIGHT , 2
In this paper, we investigate the sign changes of Fourier coefficients of half-integral weight Hecke eigenforms and give two quantitative results on the number of sign changes.
متن کاملSign Changes of Coefficients of Half Integral Weight Modular Forms
For a half integral weight modular form f we study the signs of the Fourier coefficients a(n). If f is a Hecke eigenform of level N with real Nebentypus character, and t is a fixed square-free positive integer with a(t) 6= 0, we show that for all but finitely many primes p the sequence (a(tp2m))m has infinitely many signs changes. Moreover, we prove similar (partly conditional) results for arbi...
متن کاملHecke eigenforms with rational coefficients and complex multiplication
We prove that, assuming GRH, there are only finitely many newforms with rational Fourier coefficients and complex multiplication for fixed weight up to twisting. We produce tables of such forms for weights 3 and 4, where this finiteness holds unconditionally. We also comment on geometric realizations.
متن کاملA Weak Multiplicity-one Theorem for Siegel Modular Forms
In a recent paper by Breulmann and Kohnen [BK99], the authors obtain a weak multiplicity-one result on (integral weight) Siegel-Hecke cuspidal eigenforms of degree 2, showing that such forms are completely determined by their coefficients on matrices of the form mS, where S is primitive and m is square-free. To show this, they twist Andrianov’s identity relating the Maaß-Koecher series and the ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2011