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.
