On the theory of newforms of half-integral weight (Q916692)

\textit{W. Kohnen} [J. Reine Angew. Math. 333, 32-72 (1982; Zbl 0475.10025)] established a theory of newforms of half-integral weight, paralleling the Atkin-Lehner theory of newforms of integral weight [\textit{A. O. L. Atkin} and \textit{J. Lehner}, Math. Ann. 185, 134-160 (1970; Zbl 0185.155)]. If M is an odd squarefree number, he introduced a subspace \(S^+_{k+1/2}(\Gamma_ 0(4M),\chi)\) of the space \(S_{k+1/2}(\Gamma_ 0(4M),\chi)\) of all cusp forms of weight \(k+1/2\) and character \(\chi\) on the group \(\Gamma_ 0(4M)\) in \textit{G. Shimura's} sense [Ann. Math., II. Ser. 97, 440-481 (1973; Zbl 0266.10022)], he gave an orthogonal decomposition of this \(``+\) space'' into a space of newforms and various spaces of oldforms, and he established an isomorphism of the space of newforms with the space of newforms of integral weight 2k on \(\Gamma_ 0(M)\) which is compatible with the action of Hecke operators. In the paper under review, the authors extend Kohnen's results to the full space \(S_{k+1/2}(\Gamma_ 0(4M),\chi)\).