It is reported in this paper by Zagier, as well as in Sage, that the elliptic curve $E=197A1$ has congruence number 10. (Since $E$ has prime conductor, a theorem of Ribet ensures that the congruence number is equal to the modular degree.)
If $m_E$ is the congruence number of an elliptic curve $E$, and if the newform corresponding to $E$ is $f \in S_2(\Gamma_0(N))$, then there exists another cuspidal eigenform $g \in S_2(\Gamma_0(N))$ with integral Fourier coefficients such that $f \equiv g \mod m_E$. Note that $g$ is orthogonal (with respect to the Petersson inner product) to $f$, so in particular $f \neq g$. [See the linked Zagier paper, Section 5, for equivalent formulations.]
Now we come to my confusion. One quickly checks (on LMFDB or using Sage, for instance) that there are no other cuspidal eigenforms with integral coefficients at weight $2$ and level $197$. But $m_E=10$ implies that such a form does exist, and furthermore, it should be congruent to $f$ modulo $10$. What is going on?