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