Using polar coordinates $z=re^{i\theta}$ and setting $x = r^2$ we get $$\tilde{a}_q = 1 / \left(
    2\pi R \int_0^{\infty} \frac{ \sum_{i>j}a_ia_j(i-j)^2x^{i+j-1} }{ \left( \sum a_i x^{i} \right)^2}
        x^q\mathrm{d}x
    \right).$$ Thus after substituting for $R$ we find $$\label{T map}
    T : a_q \mapsto
    \frac{
    \int_0^{\infty} \frac{ \sum_{i>j}a_ia_j(i-j)^2x^{i+j-1} }{ \left( \sum a_i x^{i} \right)^2}
        \mathrm{d}x
        }{
    (k+1) \int_0^{\infty} \frac{ \sum_{i>j}a_ia_j(i-j)^2x^{i+j-1} }{ \left( \sum a_i x^{i} \right)^3}
        x^q\mathrm{d}x}, \ q=0,1,\ldots,k.$$

By a similar computation, noting the $T_{\nu}$ map has the simpler volume form (\[nu form\]), we find $$\label{Tnu map}
    T_{\nu}: a_q \mapsto
    \left( (k+1) \int_0^{\infty} \frac{x^q \mathrm{d}x}{(1+x)^2 \sum a_i x^i} \right)^{-1}
    ,\ q=0,1,\ldots,k.$$

For the $T_K$ map the volume form is $$\omega_{G,K} = \sqrt{-1}\left( \sum a_i |z|^{2i} \right)^{-1/p} \mathrm{d}z\wedge\mathrm{d}\overline{z}$$ and we calculate as above: $$\label{TK map}
    T_K : a_q \mapsto
    \frac{
    \int_0^{\infty} \left( \sum a_i x^{i} \right)^{-2/k} \mathrm{d}x
    }{
    (k+1) \int_0^{\infty} \left( \sum a_i x^{i} \right)^{-1-2/k} x^q\mathrm{d}x
    },\ q=0,1,\ldots,k.$$

Often it is simpler still to work with ($S^1$-invariant) metrics invariant under the inversion $z \mapsto z^{-1}.$ We call such metrics *palindromic* as they are characterized as those metrics $(a_0, a_1, \ldots, a_k)$ which satisfy $$a_0 = a_k, \ a_1 = a_{k-1}, \ldots,\ a_{\lfloor k/2 \rfloor} = a_{\lceil k/2 \rceil}.$$ Thus in the palindromic 