Level 60

$$\begin{align}j_{60A}(\tau) & = j_{60E}(\tau)+\frac{1}{ j_{60E}(\tau)} = \sqrt{j_{30B}(2\tau)} \\
j_{60B}(\tau) &= \frac{(d_2\, d_{6}\, d_{10}\, d_{30})^2}{d_1\,  d_{3}\, d_{4}\, d_{5}\,d_{12}\,  d_{15}\, d_{20}\, d_{60}}\\
j_{60C}(\tau) &= \frac{a}{b} = \frac1{a}(d_6\,d_{10})^3+1 = \frac{a}{\,b^2}(d_2\,d_{30})^3-1\\
j_{60D}(\tau) &= \left(\frac{d_1\, d_{12}\, d_{15}\, d_{20}}{d_3\,  d_{4}\, d_{5}\, d_{60}}\right)\\
&=\left(\frac{d_2\, d_6\, d_{10}\, d_{30}}{d_3\,  d_{4}\, d_{5}\, d_{60}}\right)-1\\
j_{60E}(\tau) &= \frac{d_4\, d_6\, d_{20}\, d_{30} } {d_2\,  d_{10}\, d_{12}\, d_{60}} =\sqrt{j_{30D}(2\tau)}\\
j_{60F}(\tau) &= \left(\frac{d_{12}\, d_{30}}{d_6\,  d_{60}}\right)
\end{align}$$ where
$$\begin{align}
a &= d_2\,d_3\,d_5\,d_{12}\,d_{20}\,d_{30}\\
b &= d_1\,d_4\,d_6\,d_{10}\,d_{15}\,d_{60}
\end{align}$$
There are no linear relations between these functions.