  In this paper, we consider mod $\ell$ Galois representations of $\mathbb{Q}$.
In particular, we obtain an effective criterion to distinguish two semisimple
2-dimensional, odd mod $\ell$ Galois representations up to isomorphism. Serre's
conjecture (Khare-Wintenberger's theorem), Sturm's theorem, and its
modification by Kohnen are used in our proof.
