Congruence Number of 197A1 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?

 A: I am writing an answer to expand slightly on my comment. One of Zagier's definitions of the congruence number is the largest positive integer $r$ so that
there is a cusp form $g$ with integer coefficients (and not necessarily a Hecke eigenform) that is orthogonal to $f$ (under the Petersson inner product).
This is related to the problem of reducing the space of modular forms $S_{k}^{{\rm new}}(\Gamma_{0}(N))$ modulo $p$ and diagonalizing the Hecke action (modulo $p$). It is possible that there may be multiple eigenforms in $S_{k}^{{\rm new}}(\Gamma_{0}(N), \mathbb{F}_{p})$ with the same Hecke eigenvalues. The Deligne-Serre lifting lemma then implies that there are (characteristic zero) eigenforms $f$ and $g$ with coefficients in some number field and a prime ideal of residue degree $1$, $\mathfrak{p}$, so that $f \equiv g \pmod{\mathfrak{p}}$, although the coefficient fields of $f$ and $g$ can be different.
There are many ways this can occur. For example, in level $1$, every Hecke eigenform is congruent modulo $p$ to a Hecke eigenform of weight at most $p^{2}$ or so. Also, given any elliptic curve $E$, the work of Rubin and Silverberg implies the existence of infinitely many $E'$ for which the modular forms $f_{E}$ and $f_{E'}$ are congruent modulo $5$. 
