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Let $p$ be a prime, and let $E/\mathbb{Q}$ and $F/\mathbb{Q}$ be elliptic curves. Suppose the modular forms $f$ attached to $E$ and $g$ attached to $F$ have $q$-expansions that agree modulo $p$ in their coefficients up to the $n$-th term for all $n \le B$, where $B$ is the Sturm bound.

Does it then follow that the coefficients of $f$ and $g$ agree modulo $p$ for all $n$? I believe this is certainly known when $p \nmid N$ ($N$ is a product of conductors of $E$ and $F$); Proposition $4.2$ of https://wstein.org/papers/visibility_of_sha/jnt_version.pdf

Does the same property also hold when $p \mid N$? If there is a reference that proves this explicitly, I would be grateful.

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No, it does not. For example it can happen that $E$ has good reduction at $p$ while $F$ has multiplicative reduction. For example the pair of curves with isomorphic mod 17 Galois reps 3675.g1 and 47775.be.1 have $a_{13}(E) \neq a_{13}(F) \mod 17$.

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  • $\begingroup$ Thank you very much for the interesting counterexample! If E and F have the same conductor, does the claim still hold even when $p \mid N$ ? $\endgroup$ Commented Nov 20 at 14:58
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    $\begingroup$ No. It can happen that $E$ has split multiplicative reduction and $F$ has non-split, this (probably) happens in 50% of examples with $p \parallel N(E)$ and $N(F)$. Of course if $p^2 \mid N(E)$ then the $a_p(E)$ is always $0$ so... $\endgroup$ Commented Nov 20 at 20:37
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    $\begingroup$ I'm basically just copying these facts from an old paper of Kraus--Oesterlé (Sur un question de B. Mazur) by the way. It can also happen that $p^2 \mid N(E)$ and $p \parallel N(F)$ so then that's a counterexample too $\endgroup$ Commented Nov 20 at 20:41
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    $\begingroup$ Normally modular forms are normally called congruent if their Fourier coefficients agree mod $p$ for all $p \nmid N$. But anyway it is certainly still true that (2) does not imply (3). There should be counterexamples in prime level (the Sturm bound is less than the level) $\endgroup$ Commented Nov 21 at 8:54
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    $\begingroup$ I mean, if they are the same level the theorem says what the theorem says... The modular forms are equal mod p $\endgroup$ Commented Nov 22 at 15:01

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