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

### Luis Victor Dieulefait

Universitat de Barcelona, Spain### Eduardo Soto

Universitat de Barcelona, Spain

## Abstract

Let $f$ be a newform of weight $2$ on $Γ_{0}(N)$ with Fourier $q$-expansion $f(q)=q+∑_{n≥2}a_{n}q_{n}$, where $Γ_{0}(N)$ 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 $a_{p}∈{1,−1}$. We denote by $K_{f}$ the field of coefficients of $f$. Let $λ$ be a finite place in $K_{f}$ 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 $f_{′}(q)=q+∑_{n≥2}a_{n}q_{n}$ $p$-new of weight $2$ on $Γ_{0}(N)$ and a finite place $λ_{′}$ of $K_{f_{′}}$ such that $a_{p}=−a_{p}$ and the Galois representations $ρˉ _{f,λ}$ and $ρˉ _{f_{′},λ_{′}}$ are isomorphic.

## Cite this article

Luis Victor Dieulefait, Eduardo Soto, On congruences between normalized eigenforms with different sign at a Steinberg prime. Rev. Mat. Iberoam. 34 (2018), no. 1, pp. 413–421

DOI 10.4171/RMI/990