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.
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.