Langlands correspondence for weight two eigenforms over number fields

Let $K$ be a number field and let ${\mathfrak N}$ be an ideal of ${\mathcal O}_K$. Let ${\mathfrak f}$ be a non-trivial, new weight 2 complex eigenform over $K$ of level ${\mathfrak N}$. For a prime ${\mathfrak q}\nmid {\mathfrak N}$, let $\operatorname{Norm}({\mathfrak q})$ denote its absolute norm. A fake elliptic curve over $K$ is an abelian surface with the corresponding quaternionic multiplication, as intended by the source. Langlands correspondence conjecture. If $K$ has a real place, there exists an elliptic curve $E_{\mathfrak f}/K$ of conductor ${\mathfrak N}$ such that

$$\#E_{\mathfrak f}({\mathcal O}_K/{\mathfrak q})=1+\operatorname{Norm}({\mathfrak q})-{\mathfrak f}(T_{\mathfrak q})$$

for all ${\mathfrak q}\nmid {\mathfrak N}$. If $K$ is totally complex, there exists either such an elliptic curve satisfying the same formula, or a fake elliptic curve $A_{\mathfrak f}/K$ of conductor ${\mathfrak N}^2$ such that

$$\#A_{\mathfrak f}({\mathcal O}_K/{\mathfrak q})=(1+\operatorname{Norm}({\mathfrak q})-{\mathfrak f}(T_{\mathfrak q}))^2$$

for all ${\mathfrak q}\nmid {\mathfrak N}$. This is presented as a special case of a fundamental conjecture from the Langlands programme and connects automorphic eigenforms with elliptic or fake elliptic curves. The source gives no resolution status for this formulation.