Twists of newforms and pseudo-eigenvalues of \(W\)-operators
Let \(F\) be a normalized newform of level \(N\) and character \(\varepsilon\). If \(q\) is a prime dividing \(N\) and \(Q\) the \(q\)-primary factor of \(N\), then \(F| W_q=\lambda_q(F)G\) where \(G\) is a normalized newform. We call \(\lambda_q(F)\) the pseudo-eigenvalue of \(W_q\) at \(F\). In this paper we study the twist \(F_\chi\) of \(F\) by characters \(\chi\) whose conductors are powers of \(q\). We also investigate \(\lambda_q(F_\chi)\) whenever \(F_\chi\) is a newform. Without loss of generality, we may assume \(F\) \(q\)-primitive in the sense that \(F\) is not equal to a twist of a newform of level less than \(N\) by a character with \(q\)-primary conductor. If \(a(q)\), the \(q\)th Fourier coefficient of \(F\), is nonzero, then for all characters \(\chi\) with \(q\)-primary conductors, with at most one exception, \(F_\chi\) is a newform and one can express \(\lambda_q(F)\), \(\lambda_q(F_\chi)\) in terms of \(a(q)\). If \(a(q)\) is zero, then all \(F_\chi\) with conductor \(\chi\) \(q\)-primary are newforms. In this case we have an explicit formula for \(\lambda_q(F_\chi)\) if \(\text{cond}\,\chi\geq Q\) and obtain relations among \(\lambda_q(F_\chi)\)'s with \(\text{cond}\,\chi< Q\). One remarkable fact is that \(F\) is not \(q\)-primitive provided \(Q> \text{cond}\,\varepsilon_q>\sqrt Q\), where \(\varepsilon=\varepsilon_q\varepsilon_{N/q}\).
scientific article; zbMATH DE number 3575672
scientific article