$\DeclareMathOperator{\Aut}{Aut} \newcommand{\p}{\pi_1X} \newcommand{\Z}{\mathbb{Z}} \newcommand{\XX}{\tilde{X} } $ Let $X$ be a path connected and locally path connected topological space and $\rho:\p\to \Aut(A)$ a representation of the fundamental group into an abelian group $A$. We define $$H_k(X;A_\rho),$$ the homology of $X$ with coefficients in $A$ twisted by $\rho$, by taking the homology of the chain complex
$$ C_{\bullet}(\tilde{X}) \;\underset{\Z[\p]}\otimes\; A $$
where:
$\Z[\p]$ is the group ring of the fundamental group
$\tilde{X}$ is the universal cover of $X$
$C_{\bullet}(\XX)$ is the singular chain complex of $\tilde{X}$, endowed with the natural right $\Z[\p]$-module structure given by the monodromy action. That is: any $g\in \p$ acts naturally on $\XX$ as a deck transformation $g:\XX\to\XX$. Thus given a $k$-simplex $\sigma: \Delta^k\to \XX$ we define $$ \sigma \cdot g := g\circ\sigma : \Delta^k\to \XX \to \XX $$
The $\mathbb{Z}[\p]$-module structure on $A$ is given by the representation $\rho$.
I am trying to compute $H_1(X;A_\rho)$ when $X=S^1$ and $A=\Z_3$, with twist given by the nontrivial representation $\rho:\mathbb{Z}\to \Aut(\Z_3)$, i.e. the map given by $$ \rho(1) = \begin{cases} 0 \mapsto 0 \\ 1 \mapsto 2 \\ 2 \mapsto 1 \end{cases} $$
I have the geometrical picture of what's going on in my mind, but I'm having troubles with the algebra.
What I did so far:
The CW structure of $S^1$ can be pulled back to $\XX \cong \mathbb{R}$, and we obtain the cellular chain complex (as $\Z[\p]$-module) $$ 0\to \Z[\p] \to \Z[\p] \to 0, $$ and since $\Z[\p] \cong \Z[t, t^{-1}]$ the above is $$ 0\to \Z[t, t^{-1}] \to \Z[t, t^{-1}] \to 0. $$ The boundary map is given by multiplication by $t-1$.
I need help:
I can't figure out what happens to the boundary map when we apply to the above chain complex the functor $$-\;\underset{\Z[\p]}\otimes\;\Z_3$$
Can you help me with that, and thus give me the final result for this twisted homology groups?
Moreover, can this be generalized easily to an arbitrary abelian group $A$ and any representation $\rho:\p\to \Aut(A)$?