Let $f\in\mathcal{S}_k(\Gamma_1(N),\epsilon)$ is a normalized eigenform for all the Hecke operators, with character $\epsilon$, $k\geq 2$, and assume the $q$-expansion of $f$ has rational coefficients. The $L$-function $L(f,s)$ admits an Euler product of the form
$L(f,s)=\prod_{p\text{ prime}}(1-a_pp^{-s}+\epsilon(p)p^{k-1-2s})^{-1}$.
On the other hand, for a fixed prime $\ell$, we have the associated $\ell$-adic Galois representation $\rho_f:G_\mathbf{Q}\rightarrow\mathrm{GL}_2(\mathbf{Q}_\ell)$ unramified outside $\ell N$ with the characteristic polynomial of $\rho_f$ of an arithmetic Frobenius at $p\nmid\ell N$ given by $X^2-a_pX+\epsilon(p)p^{k-1}$. Bloch and Kato have defined local $L$-factors for $\rho_f$ at each prime $p$. If $p\neq\ell$, they define the local $L$-factor to be the reciprocal of the polynomial $\det(I-\rho_f(\mathrm{Frob}_p^{-1})X\vert \rho_f^{I_p})$ evaluated at $p^{-s}$, where $I_p$ is an inertia group at $p$. For $p=\ell$, the definition is similar but defined in terms of the Frobenius action of $D_{cris,\ell}(\rho_f)$.
I'm having trouble matching up the local $L$-factors for the $L$-function of $\rho_f$ with those of $f$, and I now wonder whether they coincide at all. If, for example, $p\nmid\ell N$, then, as matrices with coefficients in $\mathbf{Q}_p[X]$, we have
$I-\rho_f(\mathrm{Frob}_p^{-1})X=(-\rho_f(\mathrm{Frob}_p^{-1}))(IX-\rho_f(\mathrm{Frob}_p))$,
so, knowing what I do about the characteristic polynomial of $\rho_f(\mathrm{Frob}_p)$, I get that the Bloch-Kato local $L$-factor is $((\epsilon(p)^{-1}p^{1-k})(p^{-2s}-a_pp^{-s}+\epsilon(p)p^{k-1}))^{-1}$, which is $(1-a_pp^{1-k-s}+\epsilon(p)^{-1}p^{1-k-2s})^{-1}$. This (apparently) differs from the local $L$-factor at $p$ for $L(f,s)$.
My questions are:
(1) Have I gotten something wrong above?
(2) Should I even expect the $L$-function of the modular form to coincide with the $L$-function of the associated $\ell$-adic representation (maybe I need to dualize the representation or somehow normalize differently)?