240 and 504 showing up in the sphere spectrum

Question

It is known that $\pi_7(\mathbb S) = \mathbb Z/240$ and $\pi_{11}(\mathbb S) = \mathbb Z/504$.

Is this connected to the normalized Eisenstein series,

$E_4(\tau) = 1 + 240\sum\sigma_3(n)q^n$ and $E_6(\tau) = 1 - 504\sum\sigma_5(n)q^n$

in some direct way?

Answer 1

Something like this is explored in Hopkins: Algebraic Topology and Modular Forms (very similar looking piece at PDF page 14, internal page 296). Skimming (so take with a grain of salt), it looks like only "$c_4$" (a.k.a. $E_4$) and "$c_6$" ($E_6$) are used and (if there are any?) the remaining coefficients lie in some sort of span constructed from those two.