Let $ F$ be an imaginary quadratic field and $\mathcal{ O}$ its ring of integers. Let $ \mathfrak{ n} \subset \mathcal{ O} $ be a non-zero ideal and let $ p> 5$ be a rational inert prime in $F$ and coprime with $\mathfrak{ n}$. Let $ V$ be an irreducible finite dimensional representation of $ \bar{ \mathbb{ F}}_{ p}[{ \rm GL}_2(\mathbb{ F}_{ p^2})].$ We establish that a system of Hecke eigenvalues appearing in the cohomology with coefficients in $ V$ already lives in the cohomology with coefficients in $ \bar{ \mathbb{ F}}_{ p}\otimes det^e$ for some $ e \geq 0$; except possibly in some few cases.
Categories: math.NT