Selmer companion symmetric square representations of modular forms

Mazur–Rubin initiated the study of Selmer companion elliptic curves. Two elliptic curves over a number field K are said to be n-Selmer companion if for every quadratic twist their n-Selmer groups over K are isomorphic. Mazur–Rubin gave several sufficient conditions for two curves to be $$p^r$$ -Selmer companion for a positive power of a prime p. Jha–Majumdar–Shekhar extended this idea to the case of modular forms of weight $$k \geqslant 2$$ . In this article, we extend earlier works to give sufficient conditions which ensure that if two modular representations are $$p^r$$ -Selmer companion then their symmetric square representations are also $$p^r$$ -Selmer companion.