I'd like to ask for references on the status of modularity results for elliptic curves with CM which are not necessarily defined over $\mathbb Q$. In the case of an elliptic curve with CM defined over $\mathbb Q$ I'm aware of a nice article by Shimura, where this is explained.
Due to the fact of being an outsider in this business I would highly appreciate any hints/help towards the literature (or sharing of "common knowledge" in that field).
Maybe I should say that for me modularity means to ask for a "nice parametrization" of the elliptic curve $E$ in hand in terms of an appropriate moduli space (of (elliptic) curves). More precisely, I would also be very interested in "nice (algebraic) parametrizations" which don't match up necessarily the Hasse-Weil zeta function of $E$ with an appropriate modular form, i.e. are there "reasonable" weak forms of modularity known for elliptic curves with CM?
Thanks a lot in advance (and all my apologies in case this question is way too naive)!