For a non-negative integer $k$, define $$E_k(q)=\sum _{n=1}^{\mathrm{\infty }}{\sigma }_k(n)q^n$$ 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,
$E_1(1-\frac{1}{2^4})=3.872155809243\mathrm{e}2$
$E_3(1-\frac{1}{2^8})=2.767385314772\mathrm{e}10$
$E_7(1-\frac{1}{2^{15}})=6.725803486744\mathrm{e}39$
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.