Theory of newforms of half-integral weight (Q2352463)

Let \(N\) be squarefree and odd. The article under review concerns the theory of newforms of half-integral weight on the congruence subgroups \(\Gamma_{0} (8N)\) and \(\Gamma_{0} (16N)\). In the article, they extend the definition of the Kohnen plus space for the trivial character. Furthermore, they study newforms in the plus spaces on \(\Gamma_{0} (8N)\) and \(\Gamma_{0} (16N)\). They show that the Atkin-Lehner \(W\)-operator \(W_4\) acts as the identity operator on newforms in the space of cusp form of weight \(2k\) and level \(4N\), namely \(S_{2k}^{\mathrm{new}}(4N)\). Hence, they prove \(S_{2k}^{-}(4)=S_{2k}(4)\).