Twists of newforms
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 \(S_ k^ 0(p^{2r}M,\omega \phi)\).
scientific article; zbMATH DE number 4175030
scientific article