# Is there a converse to Vatsal's theorem on congruence of p-adic L-functions?

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 $$a_n(f)\equiv a_n(g)\:\text{mod}\:\pi^r$$.

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