Let $E_1$ and $E_2$ be elliptic curves over $\mathbb{Q}$ with good, ordinary reduction at an odd prime $p$. I'm wondering how to determine whether $E_1[p]$ and $E_2[p]$ are isomorphic $\mathbb{F}_p[G_\mathbb{Q}]$-modules.
It is my understanding that I can check this by looking for a congruence mod $p$ between the associated modular forms, but I'm not exactly sure what constitutes a congruence. I guess that it should mean that almost all of the numbers $a_\ell$ for the two curves (for $\ell\neq p$ a prime of good reduction for both $E_1$ and $E_2$) are congruent mod $p$. But this confuses me for the following reason. These $a_\ell$ are the traces of Frobenii on the $p$-adic Tate module, and I assume the reason this idea should work is that if they are (almost) all congruent mod $p$, then the $\mathbb{F}_p$-representations should be isomorphic by considerations with Chebotarev density (assuming these representations are semisimple, a fact which I think\hope is true but for which I have no reference).
If this is indeed what is meant by a congruence, then how could I check it in practice? I can look up my curves in Cremona's tables and look at as many of coefficients of the $q$-expansions of the modular forms as I want, but how many do I have to look at before I conclude that the congruence holds?
Disclaimer: I am very much new to computational stuff, so if I have said something naive or borderline ridiculous, I apologize. I'm more accustomed to working sort of...theoretically.