Eccoci.

For a non-negative integer $k$, define \[ E_k(q) = \sum\limits_{n = 1}^\infty \sigma_k(n)q^n \] where $\sigma_k(n) = \sum_{d \mid n} d^k$ is the sum of the $k$-th powers of the positive divisors of $n$.

It can be shown that, for every $k$, the series $E_k(q)$ converges for any $0 < q < 1$.

For example,
$E_1(1 - \frac{1}{2^4}) = 3.872155809243\mathrm e2$
$E_3(1 - \frac{1}{2^8}) = 2.767385314772\mathrm e10$
$E_7(1 - \frac{1}{2^{15} }) = 6.725803486744\mathrm e39$
All the above values are given in scientific notation rounded to twelve digits after the decimal point.

Find the value of $E_{15}(1 - \frac{1}{2^{25} })$.
Give the answer in scientific notation rounded to twelve digits after the decimal point.