Suppose we are given a spectrum, as explained in https://en.wikipedia.org/wiki/Spectrum_(topology) this means we are given a collection of CW complexes $\{E_k\}_{k\in \mathbb N}$ and maps $i_k:\Sigma E_k \to E_{k+1}$ inducing h-equivalences.
Now suppose we are given $X^n$ a finite CW complex of dimension $n$.
The spectrum defines a generalized cohomology theory where the cohomology in degree $d$ of a topological space $X$ is given by $$\mathcal E^d(X) = \lim_{l\to +\infty}[X_+, \Omega^l E_{d+l}]_0 $$ which I guess (?) should be also equal to $\mathcal E^d(X) = [X_+, \Omega^L E_{n+L}]_0 $ for some $L$ large enough.
Question: can we cook up, in general, a (co)chain complex depending on $X$ and on the spectrum $(C(X, \{E_k\}_k), d)$ which computes the cohomology?
This curiosity is motivated by the fact that this is possible for singular cohomology, so I was wondering if we can do it for any generalized cohomology theory.
