(This summarizes the accepted answer of Somos, and uses just the j-function and Dedekind eta function.)
Given the j-function $j(\tau)$,
$$j(\tau) = 1728J(\tau)$$
implemented in Mathematica as j(t) = 1728KleinInvariantJ[t], then Klein's septic resolvent reduces to the elegant formula,
$$y^7-7\left(\frac{-1+\sqrt{-7}}2\right)y^4-7\left(\frac{5+\sqrt{-7}}2\right)y-\sqrt[3]{j(\tau)}=0$$
where the seven roots are,
$$y_k = \frac{P_1(k)+\left(\frac{-1+\sqrt{-7}}2\right)P_2(k)}{\eta^2(\tau)}$$
with $P_1(k)$ and $P_2(k)$ for $k=1,2\dots7$ as defined by Klein (and Somos).