  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 $k$.
