The modularity conjecture for elliptic curves and Bianchi newforms

Let $K$ be an imaginary quadratic field. A Bianchi newform is a weight-$2$ automorphic form over $K$ at level $\mathcal{K}_0(\mathfrak{n})$, and an elliptic curve over $K$ has conductor ${\mathfrak a}$ when its conductor ideal is ${\mathfrak a}$. Modularity conjectures for elliptic curves over $K$.

1. If $F$ is a Bianchi newform of level $\mathcal{K}_0(\mathfrak{n})$ and weight $2$ over $K$ with rational eigenvalues, and $F$ is not a twist by a quadratic character of $G_K$ of the base-change to $K$ of an elliptic newform, then there is an elliptic curve over $K$ of conductor ${\mathfrak a}$ which is modular by $F$.
2. If $E$ is an elliptic curve over $K$ of conductor ${\mathfrak a}$ which does not have CM by an order in $K$, then $E$ is modular by some Bianchi newform $F$ of level $\mathcal{K}_0(\mathfrak{n})$ and weight $2$ over $K$.

These statements formulate the expected correspondence between suitable Bianchi newforms and elliptic curves over imaginary quadratic fields, with the stated exclusion accounting for forms arising from base change and quadratic twisting. The surrounding discussion also explains the fake elliptic curve phenomenon for exceptional rational-eigenvalue forms; no resolution is supplied here.