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 $f(q)=q+{\sum }_{n\ge 2}{a}_n{q}^n$, 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 $f^{\prime }(q)=q+{\sum }_{n\ge 2}{a}_n^{\prime }{q}^n$ $p$-new of weight $2$ on ${\Gamma }_0(N)$ and a finite place ${\lambda }^{\prime }$ of $K_{f^{\prime }}$ such that $a_p=-a_p^{\prime }$ and the Galois representations ${\bar{\rho }}_{f,\lambda }$ and ${\bar{\rho }}_{f^{\prime },{\lambda }^{\prime }}$ are isomorphic.