If you think about this question in terms of automorphic representations then it sort of becomes trivial. The space $Sk(\Gamma(N))$ can be re-interpreted as the direct sum of $\pi^{U(N)}$, where $\pi$ is running through the automorphic representations of $GL_2$ which are holomorphic of weight~$k$. Each factor is $SL(2,Z/NZ)$-invariant and I think irreducible (although my 4-year-old is sitting on my knee and wants a go on the computer, so I am being rushed and might be wrong about this), and the representation of $SL(2,Z/NZ)$ that shows up is the "type" of $\pi$. For explicit $\pi$s one will be able to explicitly determine this representation.

If you think about this question in terms of automorphic representations then it sort of becomes trivial. The space $Sk(\Gamma(N))$ can be re-interpreted as the direct sum of $\pi^{U(N)}$, where $\pi$ is running through the automorphic representations of $GL_2$ which are holomorphic of weight $k$. Each factor is $SL(2,Z/NZ)$-invariant and often irreducible but sometimes has small finite length. The representation of $SL(2,Z/NZ)$ that shows up on $\pi{^U(N)}$ is the "type" of $\pi$. For explicit $\pi$s one will be able to explicitly determine this representation.