On Hecke eigenvalues of newforms
type: Item
instance of: scholarly article
title: On Hecke eigenvalues of newforms
zbMATH Open document ID: 1090.11026
author: Winfried Kohnen
published in: Mathematische Annalen
publication date: 2004-12-29T00:00:00Z
review text: Let \(f\) and \(g\) be holomorphic cuspidal Hecke eigenforms of fixed level \(N\) and even integral weights \(k_1\) and \(k_2\), respectively which are different normalized newforms. Given \(A\in \mathbb N\), the strong multiplicity one theorem for \(\text{GL}_2\) implies that there are infinitly many primes \(p\) with \((p,A)=1\) such that \(\lambda_f(p)\neq\lambda_g(p)\), where \(\lambda_f(p)\) resp. \(\lambda_g(p)\) denote the normalized eigenvalues of \(f\) resp. \(g\). A natural question then is to ask for the smallest such prime \(p\). In this paper the author proves the following Theorem: Suppose that \(k_1,\;k_2\geq 2\) and \(k_1\neq k_2\). Let \(A\in \mathbb N\) with \(N| A\). Then there exists a prime \(p\) with \((p,A)=1\) and \(p \ll_A \max k_1,k_2\cdot| k_1-k_2|\;\) such that \(\lambda_f(p)\neq\lambda_g(p)\). The constant implied in \(\ll_A\) is effective. The proof is mainly based on the theory of modular forms modulo \(l\) due to Serre and Katz.
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 11F33
zbMATH DE Number: 2123985
zbMATH Keywords: holomorphic cuspidal Hecke eigenforms
zbMATH Keywords: multiplicity one theorem
reviewed by: Taro Horie
MaRDI profile type: Publication
full work available at URL: s00208-004-0526-1
OpenAlex ID: W2049630459
DOI: 10.1007/S00208-004-0526-1
description: scientific article; zbMATH DE number 2123985