Beta families arising from a $v_2^9$ self map on $S/(3,v_1^8)$

We show that $v_2^9$ is a permanent cycle in the 3-primary Adams-Novikov spectral sequence computing $\pi_*(S/(3,v_1^8))$, and use this to conclude that the families $\beta_{9t+3/i}$ for $i=1,2$, $\beta_{9t+6/i}$ for $i=1,2,3$, $\beta_{9t+9/i}$ for $i=1,\dots,8$, $\alpha_1\beta_{9t+3/3}$, and $\alpha_1\beta_{9t+7}$ are permanent cycles in the 3-primary Adams-Novikov spectral sequence for the sphere for all $t\geq 0$. We use a computer program by Wang to determine the additive and partial multiplicative structure of the Adams-Novikov $E_2$ page for the sphere in relevant degrees. The $i=1$ cases recover previously known results of Behrens-Pemmaraju and the second author. The results about $\beta_{9t+3/3}$, $\beta_{9t+6/3}$ and $\beta_{9t+8/9}$ were previously claimed by the second author; the computer calculations allow us to give a more direct proof. As an application, we determine the image of the Hurewicz map $\pi_*S \to \pi_*tmf$ at $p=3$.