On congruences between normalized eigenforms with different sign at a Steinberg prime

Abstract

Let $f$ be a newform of weight $2$ on $\Gamma_0(N)$ with Fourier $q$-expansion $f(q)=q+\sum_{n\geq 2} a_n q^n$, where $\Gamma_0(N)$ denotes the group of invertible matrices with integer coefficients, upper triangular mod $N$. Let $p$ be a prime dividing $N$ once, $p\parallel N$, a Steinberg prime. Then, it is well known that $a_p\in\{1,-1\}$. We denote by $K_f$ the field of coefficients of $f$. Let $\lambda$ be a finite place in $K_f$ not dividing $2p$ and assume that the mod $\lambda$ Galois representation attached to $f$ is irreducible. In this paper we will give necessary and sufficient conditions for the existence of another Hecke eigenform $f'(q)=q+\sum_{n\geq 2} a'_n q^n$ $p$-new of weight $2$ on $\Gamma_0(N)$ and a finite place $\lambda'$ of $K_{f'}$ such that $a_p=-a'_p$ and the Galois representations $\bar\rho_{f,\lambda}$ and $\bar\rho_{f',\lambda'}$ are isomorphic.