Hi,

In the paper "Lissite de la Courbe de Hecke aux points Eisenstein critiques", Bellaiche and Chenevier completely classify all reducible Galois representations coming from overconvergent eigenforms of tame level 1. They are precisely those coming from either the ordinary family of Eisenstein series, or one of the "critical" Eisenstein series

$E^{crit}_w(q) = q + \sum a_nq^n, \quad a_p=p^{k-1}, \quad a_\ell=\epsilon(\ell) + \ell^{k-1}$.

Here $w:\mathbb{Z}_p^*\rightarrow \overline{\mathbb{Q}_p^*}$ is the character $x\mapsto \epsilon(x)x^k$ where $k\geq 2$, $\epsilon$ is finite order, and $(\epsilon,k)\neq(\mathbf{1},2)$.

See Section 4, esp. Proposition 4.

In the paper "Lissite de la Courbe de Hecke aux points Eisenstein critiques", Bellaiche and Chenevier completely classify all reducible Galois representations coming from overconvergent eigenforms of tame level 1. They are precisely those coming from either the ordinary family of Eisenstein series, or one of the "critical" Eisenstein series

$E^{crit}_w(q) = q + \sum a_nq^n, \quad a_p=p^{k-1}, \quad a_\ell=\epsilon(\ell) + \ell^{k-1}$.

Here $w:\mathbb{Z}_p^*\rightarrow \overline{\mathbb{Q}_p^*}$ is the character $x\mapsto \epsilon(x)x^k$ where $k\geq 2$, $\epsilon$ is finite order, and $(\epsilon,k)\neq(\mathbf{1},2)$.

The classification is Proposition 4 of Section 4.