Cubic radial coefficient of a quadratic planar map ($\ell =\mathrm{Re}(\overline{\lambda }{g}_{21})$)

In a complex coordinate write $z^{\prime }=\lambda z+f_{20}{z}^2+f_{11}z\overline{z}+f_{02}{\overline{z}}^2$, with $|\lambda |=1$. Away from strong resonances of a planar map, the quadratic change $z=w+h_{20}{w}^2+h_{11}w\overline{w}+h_{02}{\overline{w}}^2$ has $h_{20}=f_{20}/({\lambda }^2-\lambda )$, $h_{11}=f_{11}/(1-\lambda )$ and $h_{02}=f_{02}/({\overline{\lambda }}^2-\lambda )$. Substituting and collecting $w^2\overline{w}$ gives $g_{21}=2f_{20}{h}_{11}+f_{11}{h}_{20}+f_{11}{\overline{h}}_{11}+2f_{02}{\overline{h}}_{02}$. The cubic radial coefficient is $\ell =\mathrm{Re}(\overline{\lambda }{g}_{21})$; $\ell <0$ gives a supercritical crossing when the multiplier modulus increases through one.