The usual modular parametrization says that if one takes a modular form $f$ which is a newform for $\Gamma_0(N)$ (i.e. a form in the new subspace which is a normalized eigenfunction for the Hecke operators), then one can associate to $f$ an abelian variety $A_f$ of dimension $d$, where $d$ is the degree of $K$ over $\mathbb Q$, where $K = \mathbb Q[a_n]$, being $a_n$ the Fourier coefficients of $f$ at the infinity cusp. The construction goes as follows, to $f$ one attachs the module (over $\mathbb Z$)

$$I_f = (T \in Hecke_{\mathbb Z} \quad : \quad Tf =0)$$ (sorry but the usual latex bracket didn't work in the previous formula)

Then take the quotient of $J_0(N)$ by the image of $I_f$ (i.e. $J_0(N)/I_fJ_0(N)$). My question is the following: if one considers $\mathbb Z_K$ the ring of integers of $K$ and defines $I_f$ in the same way but looks at the module generated over $\mathbb Z_F$ (which distinguishes between $f$ and its Galois conjugates), and then take the quotient of $J_0(N)$ by this ideal, do one gets an elliptic curve over $K$?

If not, I have a second question, if one considers the image of the Abel-Jacobi map for $f$, i.e. integration over all the homology of $X_0(N)$ of the form $f(z)dz$, is the image of such map a lattice in $\mathbb C$? (I would expect to get the lattice which coincides with the first question if true). I know that if you consider all the conjugates of $f$ then you get an abelian variety over $\mathbb Q$, and I wonder if it is the restriction of scalars of an elliptic curve over $K$.