# Properties of Mod $\ell^m$ Galois representation associated to modular form

(Sorry for my poor english..)

Let $$F(z)\in S_{2k}(SL_2(\mathbb{Z})$$) be a newform and $$\ell$$ be a prime larger than $$3$$. Let $$K$$ be a some number field and $$v$$ be a prime of $$K$$ over $$\ell$$. Let $$K_v$$ be a $$v$$-adic completion of $$K$$ and $$O_{K,v}$$ be a ring of integers of $$K_v$$. Let $$v=(\pi)$$. I already know that Serre and Deligne proved that there exists a Galois representation $$\rho_{F,v}$$ such that $$$$\rho_{F,v} : Gal(\overline{\mathbb{Q}}/\mathbb{Q})\to GL_2(\mathcal{O}_{K,v})$$$$ with for primes $$p\neq \ell$$, $$$$\text{Tr}(\text{Frob}_p)=\lambda_p ,\quad \det(\text{Frob}_p)=p^{2k-1}$$$$ where $$\text{Frob}_p$$ is a Frobenius element.

Let $$\rho_{F,v}^{m}$$ be a reduction of $$\rho_{F,v}$$ modulo $$(\pi^{m})$$. In other words, $$$$\rho_{F,v}^{m} : Gal(\overline{\mathbb{Q}}/\mathbb{Q})\to GL_2(\mathcal{O}_{K,v}/(\pi^m))$$$$ with for primes $$p\neq \ell$$, $$$$\text{Tr}(\text{Frob}_p)\equiv \lambda_p, \quad \det(\text{Frob}_p)\equiv p^{2k-1} \pmod{(\pi^m)}.$$$$ Does there exist $$N\in \mathbb{N}$$ such that for $$p_1\equiv p_2 \pmod{N}$$ implies that $$\rho_{F,v}^m(\text{Frob}_{p_1})\equiv \rho_{F,v}^m(\text{Frob}_{p_2})$$?

Write $$L$$ for the finite Galois extension of $$\mathbb Q$$ with Galois group $$G_{\mathbb Q}/\operatorname{Ker}\rho_{F,v}^m$$. Then $$\rho_{F,v}^m(\operatorname{Frob}_p)$$ is the identity in $$\operatorname{GL}_2(\mathcal O_{K,v}/\pi^m)$$ if and only if $$\operatorname{Frob}_p$$ is the identity in $$\operatorname{Gal}(L/\mathbb Q)$$ if and only if $$p$$ splits completely in $$L/\mathbb Q$$.
If an $$N$$ as in your question exists, then choosing it large enough so as to eliminate at most finitely many exceptions, we can arrange that there exists $$a$$ such that $$p$$ splits completely in $$L/\mathbb Q$$ if $$p\equiv a$$ modulo $$N$$. This implies that the extension $$L/\mathbb Q$$ is abelian, or equivalently that $$\rho_{F,v}^m$$ has abelian image. For $$m$$ large enough, this is never true for $$\rho_{F,v}^m$$. So the $$N$$ in your question does not exist.
The statement that $$L/\mathbb Q$$ must be abelian if $$p\equiv a$$ modulo $$N$$ implies that $$p$$ splits completely in $$L/\mathbb Q$$ is closely related to classical questions in class field theory, but is not quite stated in the usual form (for instance, it is much easier to prove that $$L/\mathbb Q$$ must be abelian if $$p\equiv a$$ modulo $$N$$ is equivalent to the statement that $$p$$ splits completely in $$L/\mathbb Q$$). Nevertheless, a complete proof can be found for instance at the following MO answer.