Diagonalizing modular forms
Let S(N,k,\(\epsilon)\) denote the space of cusp forms of weight k, level N, and character \(\epsilon\). The theory of newforms describes the structure of S(N,k,\(\epsilon)\). This paper gives a sharper result in this direction. For each prime q dividing N, an operator \(C_ q\), arising from the familiar \(U_ q\) and the Atkin-Lehner \(W_ q\) operator, is introduced as follows: \(C_ q=U_ q\) if \(\epsilon\) is not a character mod N/q; \(C_ q=U_ q+W_ q^{-1} U_ q W_ q+q^{k/2-1} W_ q^{-1}\) if \(\epsilon\) is a character mod N/q and \(ord_ q N=1\); \(C_ q=U_ q+W_ q^{-1} U_ q W_ q\) if \(\epsilon\) is a character mod N/q and \(ord_ q N>1\). Then the author proves that the space S(N,k,\(\epsilon)\) can be decomposed as a direct sum of common eigenspaces of all Hecke operators \(T_ p\), p \(| N\), and \(C_ q\), q \(\nmid N\), each of dimension one, containing a nonzero form whose eigenvalue is the pth Fourier coefficient with respect to \(T_ p\) for p \(| N\) and the qth Fourier coefficient with respect to \(C_ q\) for q \(\nmid N\). This result was first obtained by \textit{A. Pizer} [J. Algebra 83, 39-64 (1983; Zbl 0515.10020)] for the case of trivial character. The result of this paper can be extended to the space of Eisenstein series. It is also proved that the absolute eigenvalue of \(C_ q\) is equal to \(q^{(k-1)/2}\) if \(\epsilon\) is not a character mod N/q; and less than \(2q^{k/2}\) if \(\epsilon\) is a character mod N/q except possibly for the case \(ord_ q N=1\) and \(q=2\) or 3.
scientific article; zbMATH DE number 3955013