Suppose $\rho ¯:\backslash Gal(F¯/F)\to \backslash GL2(k)$ is a residual Galois representation satisfying several mild conditions, where $F$ is a number field and $k$ is a finite field with characteristics $p\ge 7$. In this work, we show that for any finite flat reduced complete intersection over $W(k)$, $R$, we can construct a deformation problem defined by local conditions imposed on some finite set of places in $F$, such that the corresponding universal deformation ring of $\rho ¯$ is $R$. It's a theorem of Wiles that if the local conditions are chosen to express restriction to deformations coming from modular forms, then the corresponding universal deformation ring is a finite flat reduced complete intersection, so our work can be regarded as a converse to Wiles' result.