A note on the minimal level of realization for a mod $\ell$ eigenvalue system

In this article we give a criterion for a mod $\ell$ eigenvalue system attached to a mod $\ell$ Katz cuspform to arise from lower level or weight. Namely, we prove the following: the eigenvalue system associated to a ring homomorphism $f:\mathbb{T }\to \overline{\mathbb{F}}_\ell$ from the Hecke algebra of level $\Gamma_1(n)$ and weight $k$ to $\overline{\mathbb{F}}_\ell$, where $\ell$ is a prime not dividing $n$ and $1\leq k \leq \ell +1$, arises from lower level or weight if there exists a prime $r$ dividing $n\ell$ such that $$\mathrm{dim}_{\overline{\mathbb{F}}_\ell} \bigcap_{p \neq r} \ker \left( T_p-f(T_p), S(n,k)_{\overline{\mathbb{F}}_\ell}\right)>1,$$ where $T_p$ is the $p$-th Hecke operator and $S(n,k)_{\overline{\mathbb{F}}_\ell}$ is the space of mod $\ell$ Katz cuspforms of level $\Gamma_1(n)$ and weight $k$.