Solve $0=(u^2+v^2)^{\frac12}\left(u\sum_{n=1}^{\infty}\sigma_3(n)ny^n\cos(t-nz)+v\sum_{n=1}^{\infty}\sigma_3(n)ny^n\sin(t+nz)\right)$ for $t$

Question

Let $t\in\left(\frac{\pi}3,\frac{2\pi}3\right)$. Let $y:=e^{-2\pi\sin t}$. Let $z:=2\pi\cos t$. Let $u:=1+240\sum_{n=1}^{\infty}\sigma_3(n)y^n\cos(nz)$. Let $v:=240\sum_{n=1}^{\infty}\sigma_3(n)y^n\sin(nz)$. Solve $0=(u^2+v^2)^{\frac12}\left(u\sum_{n=1}^{\infty}\sigma_3(n)ny^n\cos(t-nz)+v\sum_{n=1}^{\infty}\sigma_3(n)ny^n\sin(t+nz)\right).$

One solution is $t=\frac{\pi}2$.

This came up when trying to find $\max|j\Delta|$ on the fundamental domain.