$\mathcal L$-invariants and deep congruences between newforms

We study congruences modulo powers of a prime $p$ between pairs of $p$-new modular Hecke eigenforms of level $\Gamma_0(p)$ and same weight $k$. Based on explicit computations, we conjecture that every such eigenform $f$ admits a twin to which it is congruent modulo a surprisingly high power of $p$, whose exponent is close to the opposite of the valuation of the $\mathcal L$-invariant of $f$, and whose Atkin--Lehner sign is opposite to that of $f$. This is a new phenomenon that is not explained by the known results on the $p$-adic variation of eigenforms. Inspired by the global picture, we formulate a local conjecture describing congruences between semistable representations of fixed weight, varying $\mathcal L$-invariant, and opposite Atkin--Lehner signs. We give some theoretical evidence towards our conjectures.