scholarly journals Additive bases with coefficients of newforms

2018 ◽  
Vol 30 (5) ◽  
pp. 1079-1087
Author(s):  
Victor Cuauhtemoc García ◽  
Florin Nicolae
Keyword(s):  
Fourier Coefficients ◽  
Hecke Eigenform

AbstractLet {f(z)=\sum_{n=1}^{\infty}a(n)e^{2\pi inz}} be a normalized Hecke eigenform in {S_{2k}^{\mathrm{new}}(\Gamma_{0}(N))} with integer Fourier coefficients. We prove that there exists a constant {C(f\/)>0} such that any integer is a sum of at most {C(f\/)} coefficients {a(n)}. We have {C(f\/)\ll_{\varepsilon,k}N^{\frac{6k-3}{16}+\varepsilon}}.

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].

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 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.

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.

scholarly journals On the gaps between nonzero Fourier coefficients of eigenforms with CM

2017 ◽  
Vol 14 (01) ◽  
pp. 95-101
Author(s):  
Surjeet Kaushik ◽  
Narasimha Kumar
Keyword(s):  
Elliptic Curve ◽  
Fourier Coefficient ◽  
Fourier Coefficients ◽  
Short Interval ◽  
Complex Multiplication ◽  
Hecke Eigenform

Suppose [Formula: see text] is an elliptic curve over [Formula: see text] of conductor [Formula: see text] with complex multiplication (CM) by [Formula: see text], and [Formula: see text] is the corresponding cuspidal Hecke eigenform in [Formula: see text]. Then [Formula: see text]th Fourier coefficient of [Formula: see text] is nonzero in the short interval [Formula: see text] for all [Formula: see text] and for some [Formula: see text]. As a consequence, we produce infinitely many cuspidal CM eigenforms [Formula: see text] level [Formula: see text] and weight [Formula: see text] for which [Formula: see text] holds, for all [Formula: see text].

The Bombieri–Vinogradov Theorem on Higher Rank Groups and its Applications

2019 ◽  
Vol 72 (4) ◽  
pp. 928-966
Author(s):  
Yujiao Jiang ◽  
Guangshi Lü
Keyword(s):  
Fourier Coefficients ◽  
Arithmetic Function ◽  
Maass Form ◽  
Hecke Eigenforms ◽  
Hecke Eigenform ◽  
Symmetric Square ◽  
Ramanujan Conjecture ◽  
Higher Rank ◽  
Image Position ◽  
Holomorphic Hecke Eigenforms

AbstractWe study the analogue of the Bombieri–Vinogradov theorem for $\operatorname{SL}_{m}(\mathbb{Z})$ Hecke–Maass form $F(z)$. In particular, for $\operatorname{SL}_{2}(\mathbb{Z})$ holomorphic or Maass Hecke eigenforms, symmetric-square lifts of holomorphic Hecke eigenforms on $\operatorname{SL}_{2}(\mathbb{Z})$, and $\operatorname{SL}_{3}(\mathbb{Z})$ Maass Hecke eigenforms under the Ramanujan conjecture, the levels of distribution are all equal to $1/2,$ which is as strong as the Bombieri–Vinogradov theorem. As an application, we study an automorphic version of Titchmarch’s divisor problem; namely for $a\neq 0,$$$\begin{eqnarray}\mathop{\sum }_{n\leqslant x}\unicode[STIX]{x1D6EC}(n)\unicode[STIX]{x1D70C}(n)d(n-a)\ll x\log \log x,\end{eqnarray}$$ where $\unicode[STIX]{x1D70C}(n)$ are Fourier coefficients $\unicode[STIX]{x1D706}_{f}(n)$ of a holomorphic Hecke eigenform $f$ for $\operatorname{SL}_{2}(\mathbb{Z})$ or Fourier coefficients $A_{F}(n,1)$ of its symmetric-square lift $F$. Further, as a consequence, we get an asymptotic formula $$\begin{eqnarray}\mathop{\sum }_{n\leqslant x}\unicode[STIX]{x1D6EC}(n)\unicode[STIX]{x1D706}_{f}^{2}(n)d(n-a)=E_{1}(a)x\log x+O(x\log \log x),\end{eqnarray}$$ where $E_{1}(a)$ is a constant depending on $a$. Moreover, we also consider the asymptotic orthogonality of the Möbius function against the arithmetic function $\unicode[STIX]{x1D70C}(n)d(n-a)$.

scholarly journals Shintani and Shimura lifts of cusp forms on certain arithmetic groups and their applications

2017 ◽  
Vol 15 (1) ◽  
pp. 304-316
Author(s):  
SoYoung Choi ◽  
Chang Heon Kim
Keyword(s):  
Fourier Coefficients ◽  
Hecke Algebra ◽  
Cusp Forms ◽  
Arithmetic Groups ◽  
Multiplicity One ◽  
Integral Weight ◽  
Hecke Eigenform

Abstract For an odd and squarefree level N, Kohnen proved that there is a canonically defined subspace $S_{\kappa+\frac{1}{2}}^{\mathrm{n}\mathrm{e}\mathrm{w}}(N)\subset S_{\kappa+\frac{1}{2}}(N),\,\,{\text{and}}\,\,S_{\kappa+\frac{1}{2}}^{\mathrm{n}\mathrm{e}\mathrm{w}}(N)\,\,{\text{and}}\,\,S_{2k}^{\mathrm{n}\mathrm{e}\mathrm{w}}(N)$ are isomorphic as modules over the Hecke algebra. Later he gave a formula for the product $a_{g}(m)\overline{a_{g}(n)}$ of two arbitrary Fourier coefficients of a Hecke eigenform g of halfintegral weight and of level 4N in terms of certain cycle integrals of the corresponding form f of integral weight. To this end he first constructed Shimura and Shintani lifts, and then combining these lifts with the multiplicity one theorem he deduced the formula in [2, Theorem 3]. In this paper we will prove that there is a Hecke equivariant isomorphism between the spaces $S_{2k}^{+}(p)\,\,{\text{and}}\,\,\mathbb{S}_{k+\frac{1}{2}}(p).$ We will also construct Shintani and Shimura lifts for these spaces, and prove a result analogous to [2, Theorem 3].

A Variant of Lehmer’s Conjecture, II: The CM-case

2011 ◽  
Vol 63 (2) ◽  
pp. 298-326 ◽  
Author(s):  
Sanoli Gun ◽  
V. Kumar Murty
Keyword(s):  
Asymptotic Formula ◽  
Fourier Coefficient ◽  
Fourier Coefficients ◽  
Complex Multiplication ◽  
Rational Integer ◽  
Know How ◽  
Lehmer's Conjecture ◽  
Hecke Eigenform

Abstract Let f be a normalized Hecke eigenform with rational integer Fourier coefficients. It is an interesting question to know how often an integer n has a factor common with the n-th Fourier coefficient of f. It has been shown in previous papers that this happens very often. In this paper, we give an asymptotic formula for the number of integers n for which (n, a(n)) = 1, where a(n) is the n-th Fourier coefficient of a normalized Hecke eigenform f of weight 2 with rational integer Fourier coefficients and having complex multiplication.

Image Reconstruction with the Fourier Coefficients for Magnetic Induction Tomography

2020 ◽  
Vol 16 (2) ◽  
pp. 156-163
Author(s):  
Jingwen Wang ◽  
Xu Wang ◽  
Dan Yang ◽  
Kaiyang Wang
Keyword(s):  
Inverse Problem ◽  
Image Reconstruction ◽  
Magnetic Induction ◽  
Fourier Coefficients ◽  
Signal Reconstruction ◽  
Reconstruction Algorithm ◽  
Reconstruction Method ◽  
Measurement Equation ◽  
Image Reconstruction Method ◽  
Magnetic Induction Tomography

Background: Image reconstruction of magnetic induction tomography (MIT) is a typical ill-posed inverse problem, which means that the measurements are always far from enough. Thus, MIT image reconstruction results using conventional algorithms such as linear back projection and Landweber often suffer from limitations such as low resolution and blurred edges. Methods: In this paper, based on the recent finite rate of innovation (FRI) framework, a novel image reconstruction method with MIT system is presented. Results: This is achieved through modeling and sampling the MIT signals in FRI framework, resulting in a few new measurements, namely, fourier coefficients. Because each new measurement contains all the pixel position and conductivity information of the dense phase medium, the illposed inverse problem can be improved, by rebuilding the MIT measurement equation with the measurement voltage and the new measurements. Finally, a sparsity-based signal reconstruction algorithm is presented to reconstruct the original MIT image signal, by solving this new measurement equation. Conclusion: Experiments show that the proposed method has better indicators such as image error and correlation coefficient. Therefore, it is a kind of MIT image reconstruction method with high accuracy.

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