Let α₁, α₂,…, α_m be linear forms defined on ℂⁿ and $\mathcal{X}=\mathbb{C}^{n}\setminus\bigcup_{i=1}^{m}V(\alpha_{i})$ , where V(α_i)={p∈ℂⁿ:α_i(p)=0}. The coordinate ring $O_{\mathcal{X}}$ of $\mathcal{X}$ is a holonomic A_n-module, where A_n is the nth Weyl algebra and since holonomic A_n-modules have finite length, $O_{\mathcal{X}}$ has finite length. We consider a “twisted” variant of this A_n-module which is also holonomic. Define M ${}_{α}^{β}$ to be the free rank-1 ℂ[x]_α-module on the generator α^β (thought of as a multivalued function), where $\alpha^{\beta}=\alpha_{1}^{\beta_{1}},\ldots,\alpha_{m}^{\beta_{m}}$ and the multi-index β=(β₁,…, β_m)∈ℂ^m. Our main result is the computation of the number of decomposition factors of M ${}_{α}^{β}$ and their description when n=2.