Twists of newforms
type: Item
instance of: scholarly article
title: Twists of newforms
zbMATH Open document ID: 0714.11028
DOI: 10.1016/0022-314X(90)90119-C
author: Hiroaki Hijikata
author: Arnold K. Pizer
author: Thomas R. Shemanske
published in: Journal of Number Theory
publication date: 1990-01-01T00:00:00Z
review text: Let \(S_ k^ 0(N,\psi)\) denote the space of newforms of weight \(k\) and character \(\psi\) on the congruence subgroup \(\Gamma_ 0(N)\) of level \(N\). If \(F(\tau)\) is a newform in \(S_ k^ 0(N,\psi)\) and \(\chi\) is a Dirichlet character, then \(F_{\chi}(\tau)\), the twist of \(F(\tau)\) by \(\chi\), is a cusp form of weight \(k\) and character \(\psi \chi^ 2\) on \(\Gamma_ 0(N')\) for some \(N'\). When the conductor \(f(\chi)\) of \(\chi\) is relatively prime to \(N\), then the behaviour of \(F_{\chi}(\tau)\) is rather easy to study. When \(f(\chi)\) is not relative prime to \(N\), the situation is more complicated and has been studied by \textit{A. O. L. Atkin} and \textit{W.-C. W. Li} [Invent. Math. 48, 221--243 (1978; Zbl 0369.10016)]. In the present paper the authors study the latter case and related questions. However, they stress the differences between their point of view, methods, results and those of Atkin and Li. We briefly observe the contents of the paper. Let \(S_ k(N,\psi)\) denote the space of all cusp forms of weight \(k\) and character \(\psi\) on \(\Gamma_ 0(N)\). An important result from the theory of newforms is the decomposition \[ S_ k(N,\psi)\cong \oplus_{f(\psi)| a| N}\delta (N/a)S_ k^ 0(a,\psi), \] where \(\delta\) is the multiplicity. Denote by \(\text{tr}_{N,\psi}T_ k(n)\) the trace of \(T_ k(n)\) acting on the space \(S_ k(N,\psi)\). The paper quotes a formula for \(\text{tr}_{N,\psi}T_ k(n)\) from the first author's paper [J. Math. Soc. Japan 26, 56--82 (1974; Zbl 0266.12009)], with appending the explicit evaluation of the contribution of the parabolic terms to the trace formula. Then they give isomorphisms between the twists of various spaces of newforms and also decompositions of spaces of newforms as direct sums of twists of other spaces of newforms. In several places they compare their results with those of Atkin-Li. The paper ends with giving a study of the special space \(S_ k^ 0(p^{2r}M,\omega \phi)\).
reviewed by: Michio Ozeki
Mathematics Subject Classification ID: 11F11
Mathematics Subject Classification ID: 11F25
Mathematics Subject Classification ID: 11F27
zbMATH DE Number: 4175030
zbMATH Keywords: Hecke operator
zbMATH Keywords: Dirichlet character
zbMATH Keywords: cusp form
zbMATH Keywords: trace formula
zbMATH Keywords: twists
zbMATH Keywords: spaces of newforms
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4175030