The function field of $X_0(3^n)$ is generated by
$x_1,\ldots,x_{k-1}$
with the cubic relations
$(z_j^3-1) (x_{j-1}^3-1) = 1$
where
$z_j = \frac{x_j+2}{x_j-1}.$
Here $x_1 = \xi(\tau) = 1 + (\eta(\tau)/\eta(9\tau))^3/3$
is a rational parameter for $X_0(9)$, and $x_j=\xi(3^{j-1}\tau)$.
