(*This summarizes the accepted answer of Somos, and uses just the j-function and Dedekind eta function*.) 

Given the [*j-function*][1] $j(\tau)$,

$$j(\tau) = 1728J(\tau)$$

implemented in *Mathematica* as j(t) = 1728KleinInvariantJ[t], then *Klein's septic resolvent* reduces to the elegant formula,

$$y^7-7\left(\tfrac{-1+\sqrt{-7}}2\right)y^4-7\left(\tfrac{5+\sqrt{-7}}2\right)y-\sqrt[3]{j(\tau)}=0$$

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).


  [1]: http://mathworld.wolfram.com/j-Function.html