Let $E$ be an elliptic curve over a number field.

When $E$ has no CM and is a $\mathbb Q$-curve (i.e. it is $\overline{\mathbb Q}$-isogenous to all of its conjugates), it is nowadays known that $E$ is a factor up to isogeny of $J_1(N)$. In a nice paper X. Guitart and J. Quer define a subclass of $\mathbb Q$-curves, called strongly modular curves, which enjoy the property that their $L$-function is a product of $L$-functions of newforms. Thus, the $L$-function of these curves can be analitically continued to $\mathbb C$.

If $E$ has CM, the $L$-function of $E$ is known to admit an analytic continuation to $\mathbb C$ because it coincides with a product of $L$-functions of Hecke characters.

My first question is: are $L$-functions of elliptic curves with CM also products of $L$-functions of (classical) newforms? It is not clear at all to me what is the connection between $L$-functions of cuspforms and $L$-functions of Hecke characters.

My second question is: what is the state of art of the problem? Namely, is there any other class of elliptic curves whose $L$-function is known to have an analytic continuation?