Dimension formulas of modular form spaces with character for Fricke groups

We first obtain the dimension formulas for the spaces of holomorphic modular forms with character for the Fricke group ${\mathrm{\Gamma }}_0^{+}(N)$, then that for ${\mathrm{\Gamma }}_0^{\ast }(N)$ with all Atkin-Lehner involutions added in a particular case.