Let N be a positive integer and K be a number field. Suppose that \(f_1, f_2\in S_k(\Gamma _0(N))\) are two newforms such that their residual Galois representations at 2 are isomorphic. Let \(\omega _2:G_{\mathbb {Q}}\rightarrow {\mathbb {Z}}_2^*\) be the 2-adic cyclotomic character. Then, under suitable hypotheses, we have shown that for every quadratic character \(\chi \) of K and each critical twist j, the residual Greenberg 2-Selmer groups of \(f_1\chi \omega _p^{-j}\) and \(f_2\chi \omega _p^{-j}\) over K are isomorphic. This generalizes the corresponding result of Mazur–Rubin on 2-Selmer companion elliptic curves. Conversely, if the difference of the residual Greenberg (respectively Bloch–Kato) 2-Selmer ranks of \(f_1\chi \) and \(f_2\chi \) is bounded independent of every quadratic character \(\chi \) of K, then under suitable hypotheses we have shown that the residual Galois representations at 2 of \(f_1\) and \(f_2\) are isomorphic as \(G_K\) -modules. The corresponding result for elliptic curves was a conjecture of Mazur–Rubin, which was proved by M. Yu.
