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Let $f=\sum_n a_n(f) q^n$ and $g=\sum_n a_n(g) q^n$ be normalized (cuspidal) newforms whose Fourier coefficients are contained in the p-adic field K for which the uniformizer of $\mathcal{O}_K$ is denoted by $\pi$. Suppose that the Fourier coefficients of $f$ and $g$ are congruent modulo $\pi^r$ i.e. $a_n(f)\equiv a_n(g)\:\text{mod}\:\pi^r$. Vatsal in his paper on canonical periods (Vatsal, Vinayak. "Canonical periods and congruence formulae." Duke mathematical journal 98.2 (1999): 397.) established that under further hypotheses, there is a congruence of the associated p-adic L-functions $L_p(f,s)\equiv L_p(g,s)\:\text{mod}\:\pi^r$.

Is there a converse to this result, namely if $f$ and $g$ are normalized (cuspidal) newforms whose Fourier coefficients are contained in $K$ and that $L_p(f,s)\equiv L_p(g,s)\:\text{mod}\:\pi^r$, then it should follow that $a_n(f)\equiv a_n(g)\:\text{mod}\:\pi^r$.

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    $\begingroup$ I think that the answer is no, if you don't consider twists by Dirichlet characters. $\endgroup$ – Emmanuel Lecouturier Jun 12 at 4:45

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