Since 2008 we have the following remarkable correspondence:



> 
> Odd irreducible 2-dim Galois repn $\longleftrightarrow$ weight 1
>  newforms
> 
> 
> 


*note: all Galois representations in this question are ment to be continuous complex linear representations.*


That this is a bijection is the consequence of three deep results.


* Serre-Deligne: for each weight 1 newform there is an irreducible 2-dim Galois representation.
* Weil-Langlands: the converse is true if the Artin conjecture holds for those representations.
* Khare-Wintenberger: the Artin conjecture holds for odd irreducible 2-dim Galois representation (from Serre's modularity conjecture).


Now, we also famously have (Wiles-Breuil-Conrad-Diamond-Taylor):



> 
> Elliptic curves over $\mathbb{Q}$ $\longrightarrow$ weight 2
>  newforms
> 
> 
> 


My question is, **what is known/conjectured on the converse direction?** This is, does every newform of weight 2 arises from an elliptic curve over $\mathbb{Q}$? (Note that this correspondence can only hold up to isogeny) Are there any other source for newforms of this weight? How can we tell apart those that came from elliptic curves?



Converse to Modularity I: weight 2 newforms
This should be a comment, but is getting too long.


First, unless you are using the strange convention that Galois representations have by definition complex coefficients, odd, irreducible 2-dimensional Galois representations do not correspond to weight 1 newforms. Only a very small subset of the former set corresponds to the latter one, namely the one with values in $\operatorname{GL}\_2(\mathbb C)$ or with finite image.


Second, you exchanged the contributions of Weil-Langlands and Deligne-Serre.


Third, the newforms whose $L$-function is the $L$-function of an elliptic curve are exactly those with integral coefficients. This is elementary (at least if one disregards Euler factors at places of bad reduction, in order to conclude for all Euler factors, one needs Faltings's theorem): just take the quotient of the Jacobian of the modular curve given by the Hecke action on $f$. The other newforms of weight 2 are attached to modular abelian varieties (the dimension of the abelian variety being the degree of the field generated by the coefficients of $f$ over $\mathbb Q$). This is a classical result of G.Shimura.


