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Since it hasn't yet been mentioned, $x^4+2x+2$ is an explicit polynomial with Galois group $S_4$ over $\mathbf{Q}_2$, so $K = \mathbf{Q}_2[x]/(x^4+2x+2)$ is a degree $4$ extension with no quadratic subfields.
An analogy: the global Langlands program for $\mathrm{GL}(2)$ says quite a lot about the arithmetic of torsion free congruence subgroups $\Gamma$ of $\mathrm{GL}_2(\mathbf{Z})$, but I don't think it has had very much to say about these groups as abstract groups, since they are, in the end, just free groups.
Hecke operators relate to the local structure of the group. But $U(n,n)/K$ over (say) a prime that splits in $K$ looks like $\mathrm{GL}(2n)$, so the Hecke theory looks like that of $\mathrm{GL}(2n)$ for such primes.
Your comment is pretty vague. But you should really just learn the recipe of how all these things connect (Satake parameters, Langlands dual group, L-functions)are connected and then the conjectural story is just Lie theory. For example, the dual group of $U(n,n)$ is something close to $GL(2n)$ semi-direct product with $\mathrm{Gal}(K/\mathbf{\mathbf{Q}})$ acting by conjugate transpose. So $L$-functions related to representations of this group. Similarly, Satake paremeters are related to elements of this group. (When $n = 1$ this automorphism is inner, which is why things are simpler.)
For $n = 2$, these degree $4$ and $5$ $L$-functions are the classical ones which are known to have analytic continuations. (Note $\mathrm{Gspin}(5) \simeq \mathrm{GSp}(4)$ by some accidental automorphism.) But there are other $L$-functions, for example, $\mathrm{GSp}(4)$ has a (unique) irreducible representation $\rho$ of dimension $91$, so there should also be an $L$-function $L(\pi,\rho,s)$ for a Siegel modular form. (It exists, but its analytic continuation is not known.)
It is exactly an example of this. If $G = \mathrm{GSp}(2n)$, then the dual group is $\mathrm{Gspin}(2n+1)$. The latter group has many representations, including a "spin" representation of dimension $2^n$ and a "standard" representation of dimension $2n+1$, so specifying that representation is part of the data defining an $L$-function for $\mathrm{GSp}(2n)$.
@Heavensfall, I originally read your original question as giving the size of $X(\mathbf{F}_{p^n})$ for all $p$ between $1$ and $N$. Fortunately, I think my original answer contains an answer to your actual question.
