 Vanishing harmonic moments of a polynomial conformal disc map

If a degree-$N$ polynomial $f(\zeta )={\sum }_{j=1}^N{a}_j{\zeta }^j$ maps the unit disc conformally onto $D$, Green's theorem expresses $M_n={\iint }_D{z}^{n}dxdy$ as the residue of $f^n{f}^{\prime }{\sum }_j{\bar{a}}_j{\zeta }^{-j}$ times $\pi$. Its lowest possible power is $n-N$, so no residue occurs when $n\ge N$. For a quadratic map, $M_0=\pi (|a_1{|}^2+2|a_2{|}^2)$ and $M_1=\pi a_1^2{\bar{a}}_2$. The first moment can be complex; the zeroth is area.