On the Petersson norm of a Siegel-Hecke Eigenform of degree two in the Maass space.

1985 ◽  
Vol 1985 (357) ◽  
pp. 96-100 ◽  
Keyword(s):  
Hecke Eigenform ◽  
Petersson Norm

scholarly journals Symmetric power functoriality for holomorphic modular forms, II

2021 ◽  
Author(s):  
James Newton ◽  
Jack A. Thorne
Keyword(s):  
Modular Forms ◽  
Complex Multiplication ◽  
Symmetric Power ◽  
Hecke Eigenform ◽  
Holomorphic Modular Forms

AbstractLet $f$ f be a cuspidal Hecke eigenform without complex multiplication. We prove the automorphy of the symmetric power lifting $\operatorname{Sym}^{n} f$ Sym n f for every $n \geq 1$ n ≥ 1 .

Sign changes of Fourier coefficients of cusp forms supported on prime power indices

2014 ◽  
Vol 10 (08) ◽  
pp. 1921-1927 ◽  
Author(s):  
Winfried Kohnen ◽  
Yves Martin
Keyword(s):  
Positive Integer ◽  
Prime Power ◽  
Fourier Coefficients ◽  
Cusp Forms ◽  
Power Indices ◽  
Hecke Operator ◽  
Sign Changes ◽  
Integral Weight ◽  
Hecke Eigenform ◽  
Almost All

Let f be an even integral weight, normalized, cuspidal Hecke eigenform over SL2(ℤ) with Fourier coefficients a(n). Let j be a positive integer. We prove that for almost all primes p the sequence (a(pjn))n≥0 has infinitely many sign changes. We also obtain a similar result for any cusp form with real Fourier coefficients that provide the characteristic polynomial of some generalized Hecke operator is irreducible over ℚ.

scholarly journals Fourier coefficients of the overconvergent generalized eigenform associated to a CM form

2020 ◽  
Vol 16 (06) ◽  
pp. 1185-1197
Author(s):  
Chi-Yun Hsu
Keyword(s):  
Modular Form ◽  
Fourier Coefficients ◽  
Complex Multiplication ◽  
Hecke Eigenvalues ◽  
Hecke Eigenform ◽  
Critical Slope

Let [Formula: see text] be a modular form with complex multiplication. If [Formula: see text] has critical slope, then Coleman’s classicality theorem implies that there is a [Formula: see text]-adic overconvergent generalized Hecke eigenform with the same Hecke eigenvalues as [Formula: see text]. We give a formula for the Fourier coefficients of this generalized Hecke eigenform. We also investigate the dimension of the generalized Hecke eigenspace of [Formula: see text]-adic overconvergent forms containing [Formula: see text].

scholarly journals Period polynomials and congruences for Hecke algebras

1999 ◽  
Vol 42 (2) ◽  
pp. 217-224 ◽  
Author(s):  
Winfried Hohnen
Keyword(s):  
Modular Group ◽  
Hecke Algebras ◽  
Hecke Operators ◽  
Hecke Operator ◽  
Hecke Eigenform ◽  
Image Position

Using the Eichler-Shimura isomorphism and the action of the Hecke operator T2 on period polynomials, we shall give a simple and new proof of the following result (implicitly contained in the literature): let f be a normalized Hecke eigenform of weight k with respect to the full modular group with eigenvalues λp under the usual Hecke operators Tp (p a prime). Let Kf be the field generated over Q by the λp for all p. Let p be a prime of Kf lying above 5. Then

ODD VALUES OF FOURIER COEFFICIENTS OF CERTAIN MODULAR FORMS

2007 ◽  
Vol 03 (03) ◽  
pp. 455-470 ◽  
Author(s):  
M. RAM MURTY ◽  
V. KUMAR MURTY
Keyword(s):  
Modular Forms ◽  
Fourier Coefficients ◽  
Hecke Operator ◽  
Hecke Eigenform ◽  
Relationship Of ◽  
The Relationship

Let f be a normalized Hecke eigenform of weight k ≥ 4 on Γ0(N). Let λf(n) denote the eigenvalue of the nth Hecke operator acting on f. We show that the number of n such that λf(n) takes a given value coprime to 2, is finite. We also treat the case of levels 2aN0 with a arbitrary and N0 = 1, 3, 5, 15 and 17. We discuss the relationship of these results to the classical conjecture of Lang and Trotter.

scholarly journals Power Moments of the Riesz Mean Error Term of Symmetric Square L-Function in Short Intervals

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 2036
Author(s):  
Rui Zhang ◽  
Xue Han ◽  
Deyu Zhang
Keyword(s):  
Error Term ◽  
Asymptotic Formulas ◽  
Short Intervals ◽  
Mean Error ◽  
Riesz Mean ◽  
Hecke Eigenform ◽  
Symmetric Square ◽  
Power Moments

Let f(z) be a holomorphic Hecke eigenform of weight k with respect to SL(2,Z) and let L(s,sym2f)=∑n=1∞cnn−s,ℜs>1 denote the symmetric square L-function of f. In this paper, we consider the Riesz mean of the form Dρ(x;sym2f)=L(0,sym2f)Γ(ρ+1)xρ+Δρ(x;sym2f) and derive the asymptotic formulas for ∫T−HT+HΔρk(x;sym2f)dx, when k≥3.

scholarly journals Twisted Maaß–Koecher series and spinor zeta functions

1999 ◽  
Vol 155 ◽  
pp. 153-160 ◽  
Author(s):  
Stefan Breulmann ◽  
Winfried Kohnen
Keyword(s):  
Fourier Coefficients ◽  
Zeta Functions ◽  
Positive Definite ◽  
Integral Weight ◽  
Genus 2 ◽  
Hecke Eigenform ◽  
Positive Integers

AbstractIt is shown that a Siegel-Hecke eigenform of integral weight k and genus 2 is uniquely determined by its Fourier coefficients indexed by nT where T runs over all half-integral positive definite primitive matrices of size 2 and n over all squarefree positive integers. The proof uses analytic arguments involving Koecher-Maaß series and spinor zeta functions.

Remarks on the rightmost critical value of the triple product L-function

2021 ◽  
pp. 1-16
Author(s):  
Kengo Fukunaga ◽  
Kohta Gejima
Keyword(s):  
Critical Value ◽  
Product Formula ◽  
Orthogonal Basis ◽  
Triple Product ◽  
Explicit Formulas ◽  
Arithmetic Expression ◽  
Weighted Averages ◽  
Hecke Eigenforms ◽  
Hecke Eigenform ◽  
The Individual

Let [Formula: see text] be a normalized cuspidal Hecke eigenform. We give explicit formulas for weighted averages of the rightmost critical values of triple product [Formula: see text]-functions [Formula: see text], where [Formula: see text] and [Formula: see text] run over an orthogonal basis of [Formula: see text] consisting of normalized cuspidal Hecke eigenforms. Those explicit formulas provide us an arithmetic expression of the rightmost critical value of the individual triple product [Formula: see text]-functions.

scholarly journals A remark on the Shimura correspondence

1988 ◽  
Vol 30 (3) ◽  
pp. 285-291 ◽  
Author(s):  
Winfried Kohnen
Keyword(s):  
Fourier Coefficients ◽  
Central Point ◽  
Modular Curve ◽  
Special Values ◽  
Half Integral Weight ◽  
Integral Weight ◽  
Hecke Eigenform ◽  
Special Case ◽  
Shimura Correspondence

In [4] an identity is given which relates the product of two Fourier coefficients of a Hecke eigenform g of half-integral weight and level 4N with N odd and squarefree to the integral of a Hecke eigenform f of even integral weight associated to g under the Shimura correspondence along a geodesic period on the modular curve X0(N) This formula contains as a special case a refinement of a result of Waldspurger [6] about special values of L-series attached to f at the central point.

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