Let $f$ be a modular form of weight $k>0$ on $\Gamma_1(N)$ that is an eigenvector for all the Hecke operators $T_\ell$, for $\ell$ prime, and with Nebentype $\epsilon$. If $\ell$ is a prime not dividing $N$, set $\iota_\ell(f)(z):=f(\ell z)$. As it is well known, $f$ and $\iota_\ell(f)$ are distinct modular forms of weight $k$ on $\Gamma_1(N\ell)$ that span a two-dimensional space denoted by $V_f$. It is also well known that any Hecke operator $T_p$, for $p$ a prime other than $\ell$, acts on $V_f$ as a scalar, and that the characteristic polynomial of the linear action of $T_\ell$ (often denoted by $U_\ell$ in this context) on $V_f$ is $X^2-a_\ell X +\epsilon(\ell)\ell^{k-1}$, where $a_\ell$ is the $\ell$--th eigenvalue of our original $f\in M_k(\Gamma_1(N))$. A possible proof of this fact is a simple computation based on the explicit formulas for the effect of Hecke operators on $q$--expansions.

On the other hand, it is also well known that if $\lambda$ is a finite prime of the number field attached to $f$, and $\rho_\lambda$ is the two dimensional, $\lambda$--adic Galois representation of $Q$ attached to $f$, then the "Hecke polynomial" mentioned in the previous paragraph coincide with the characteristic ("Galois") polynomial of $\rho_\lambda$ at a Frobenius element at $\ell$, when $\lambda$ does not divide $\ell$.

Question: is there a conceptual proof of the equality between these two polynomials? Can you point a reference where this fact is explained? thanks.