Classify weight-one Eisenstein–cuspidal product identities

Determine all full-level Hilbert Hecke eigenform product identities consisting of a weight-one Eisenstein eigenform and a cuspidal Hecke eigenform of positive parallel integral weight over totally real fields of degree at least two.

Background

The paper classifies full-level Hilbert Hecke eigenform products over totally real fields of degree at least two. It proves complete nonexistence or uniqueness results for products of two Eisenstein eigenforms, for products of an Eisenstein eigenform of weight at least two with a cuspidal eigenform, and for products of two cuspidal eigenforms.

The remaining classification range is a weight-one Eisenstein factor multiplied by a cuspidal factor. The paper establishes one example over $\mathbb Q(\sqrt3)$, namely $h_5=12E_1(1,\varepsilon)\cdot h_4$, where $\varepsilon$ is the nontrivial narrow class character and $h_4\in S_4(1)$. However, the summary table explicitly records the general $E_1\times C_\ell$ case as unresolved.