On twisting operators and newforms of half-integral weight. III: Subspace corresponding to very-newforms (Q2769539)

[For Part II, cf. Nagoya Math. J. 149, 117-171 (1998; Zbl 1016.11505).]NEWLINENEWLINENEWLINELet \(k\) and \(N\) be positive integers and \(\chi\) an even quadratic character modulo \(N\). Suppose that \(N=4M\) with an odd positive integer \(M\). Let \(S(k+1/2,N,\chi)\) be the Kohnen space and \( D(k+1/2,N,\chi)\) its subspace of ``oldforms''. We denote by \( N(k+1/2,N,\chi)\) the orthogonal complement of \( D (k+1/2,N,\chi)\) in \(S(k+1/2,N,\chi)\) with respect to the Petersson inner product. Let \(\prod\) be the set of all odd prime divisors \(p\) of \(N\) such that \(p^{2}\mid N\). Since the subspace \( N(k+1/2,N,\chi)\) is fixed by the twisting operator \(R_{p}: \sum_{n\geq 1} a(n)e(nz)\mapsto \sum_{n\geq 1} a(n)(\frac{n}{p})e(nz)\) for any \(p\in \prod\), this subspace can be decomposed into common eigensubspaces \( N^{\varphi,\kappa}(k+1/2,N,\chi)\) on these twisting operators where \(\kappa\) runs over all maps from \(\prod\) to \({\pm 1}\). We call these subspaces the spaces of newforms for Kohnen spaces. There exists an embedding \(\dagger\) as a Hecke modulo from \( N^{\phi,\kappa}(k+1/2,N,\chi)\) into \(S^{0}(2k,N/4)\) which is the space of newforms of weight \(2k\) and of level \(N/4\). We know that the image of \(\dagger\) contains lifting \(G\mid R_{\psi}\) of cusp forms of lower level \(M'\) \((0<M'|N/4, M'<N/4)\) with twisting operators \(R_{\psi}\). A newform \(F\) of weight \(2k\) is called a very-newform if \(F\) is orthogonal to all these liftings \(G\mid R_{\psi}\). The aim of this paper is to find the subspace of \( N^{\varphi,\kappa}(k+1/2,N,\chi)\) which corresponds by \(\dagger \) to the space of very-newforms.