Remark on the dimension of Kohnen's spaces of half-integral weight with square free level
For an odd integer \(N\), a Dirichlet character \(\chi\) modulo \(N\), and an integer \(k\geq 0\), consider the space \(S_{k+1/2} (4N,\chi_\varepsilon)\) of cusp forms of half-integral weight \(k+{1\over 2}\), level \(4N\) and character \(\chi_\varepsilon\), where \(\varepsilon= \chi(-1)\) and \(\chi_\varepsilon= ({4 \varepsilon\over \cdot})\chi\). Kohnen's subspace \(S_{k+1/2} (N,\chi)\) consists of all those \(f\) in \(S_{k+1/2} (4N,\chi_\varepsilon)\) whose Fourier expansion is of the form \(f(z)= \sum_na(n) e^{2\pi inz}\) where the summation is restricted by the condition that \(\varepsilon(-1)^kn \equiv 0\) or 1 \(\pmod 4\). The author uses a result of \textit{H. Cohn} and \textit{J. Oesterlé} [Lect. Notes Math. 627, 69-78 (1977; Zbl 0371.10020)] on the dimension of \(S_{k+ 1/2}(4N, \chi_\varepsilon)\) to prove that \(\dim S_{k+1/2} (N,\chi)= \dim S_{2k} (N,\chi^2)\) if \(N\) is square-free and \(k\geq 2\). The difference \(\dim S_{k+1/2} (N,\chi)- {1\over 3}\dim S_{k+1/2} (4N,\chi_\varepsilon)\) is expressed in elementary terms.
scientific article; zbMATH DE number 1782993