I want to know how to construct this modular Galois representation:
The $\bmod p$ representation is semi-simple.
For any lattice $T$ , $\dfrac{T}{p^{3}T}$ does not have a $G$-stable cyclic subgroup of order $p^3$ (i.e., there does not exist the congruence equation $a_\ell=\ell^{s}+\ell^{t} \bmod p^3$ (for prime $\ell\neq p$)