Consider the $D_{\mathbb{A}^1}$-module $M:=D_{\mathbb{A}^1}/(x)$, and the map $f:z\mapsto z^k$. I want to know $f^*(M)$. I believe it only has a single non-zero cohomology, namely in degree $0$, which is equal to $$\mathbb{C}[z]\otimes_{\mathbb{C}[z^k]}(D/(x)),$$ with connection $$\partial(z^q\otimes \partial^w)=qz^{q-1}\otimes \partial^w+kz^{q+k-1}\otimes \partial^{w+1}.$$ Now by general theory this should be holonomic, and in particular finitely generated. However, I have some trouble finding a generating set. For example, it is not clear to me how the elements $1\otimes \partial^{w}, w=1,2,\dots,$ can be generated from a finite generating set.

Any help or hints would be appreciated.