In terms of the data of which you speak, you can use asymptotics of the Fourier coefficients on the modular form side and the Hasse bound on the elliptic curve side. Specifically, if $f$ is a normalized newform of weight $k$ (an integer $\geq2$, I'm not enough of an expert on this to know if more general $k$ are allowed), then $$\sum_{n\leq X}|a_n|^2/n^{k-1}=c_fX+O(X^{3/5})$$ (see section 14.9 of Iwaniec–Kowalski), but, on the other hand, the Hasse bound for an elliptic curve states that $$|a_p|<2\sqrt{p}$$ (see this mathoverflow answer to find out what this implies about the $a_n$ of the putative cusp form attached to the elliptic curve). Given this bound, $k$ must be at most 2.
Update: From reading the mathscinet review of an old paper of Selberg's, it looks like the asymptotic on the Fourier coefficients (I guess originally due to Rankin) works for any weight which is a positive real number (but the language in the review is rather old-fashioned and I don't have access to the paper right now). I'll take a look probably in a couple of days.
