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 curves are exactly those  with integral coefficients. This is elementary: 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.