# Conductor of Galois representation attached to newform

(Sorry for poor my english skill..)

Let $$k$$ and $$N$$ be positive integers and $$\chi$$ be a Dirichlet character modulo $$N$$. Let $$F$$ be a newform with number field $$K_{F}$$. (All coefficients of $$F$$ in $$K_{F}$$.) Let $$\ell$$ be a prime and let $$\lambda$$ of $$\mathcal{O}_{K_{F}}$$ be a maximal ideal lying over $$\ell$$. By theorem 9.6.5 in "A first course in Modular forms - Diamond and Shurman", there is an irreducible 2-dimensional Galois representation $$\begin{equation} \rho_{F,\lambda} : G_{\mathbb{Q}} \to GL_2(K_{F,\lambda}). \end{equation}$$ Someone told me that the conductor of $$\rho_{F,\lambda}$$ is same as the level of $$F$$ i.e. the conductor of $$\rho_{F,\lambda}$$ is $$N$$. Also he said that this fact is in the Carayol's paper, however I couldn't find it.

Is the fact true? If it is true, I would appreciate your reference.

In fact much more than the equality of conductor is true: the local Galois representation $$\rho_{F,\lambda}|G_{\mathbb Q_{p}}$$ obtained by restricting $$\rho_{F,\lambda}$$ to the decomposition group at $$p$$ corresponds in a precise way to the local automorphic representation $$\pi(F)_{p}$$. This is the so-called local-global compatibility property of the Langlands reciprocity conjectures.
Sur les représentations $$l$$-adiques associées aux formes modulaires de Hilbert Carayol, Henri. Annales scientifiques de l'É.N.S tome 19 n°3 (1986) page 409-468.
*To be precise, Carayol's work goes through a compatible system of Galois representation, if you want to relate the single Galois representation $$\rho_{F,\lambda}$$ directly to $$\pi(F)_\ell$$ (so if you want to the power of $$\ell$$ appearing in the conductor and level without appealing to the compatible system attached to $$F$$), then you need a result of Takeshi Saito (Inventiones mathematicae,1997).