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

Abstract

Let f be a newform of weight 2 on Γ0(N) with Fourier q-expansion f(q)=q+∑n≥2anqn, where Γ0(N) denotes the group of invertible matrices with integer coefficients, upper triangular mod N. Let p be a prime dividing N once, p∥N, a Steinberg prime. Then, it is well known that ap∈{1,−1}. We denote by Kf the field of coefficients of f. Let λ be a finite place in Kf not dividing 2p and assume that the mod λ 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+∑n≥2a′nqn p-new of weight 2 on Γ0(N) and a finite place λ′ of Kf′ such that ap=−a′p and the Galois representations ρ¯f,λ and ρ¯f′,λ′ are isomorphic.