Let $N \geq 1$ be an integer and let $p$ be a prime such that $p \nmid N$. Let $J_0(N)_{\mathbb{F}_p}$ denote the special fibre at $p$ of the Jacobian $J_0(N)$, let $F_p \in \operatorname{End}(J_0(N)_{\mathbb{F}_p})$ be the $p$-power Frobenius endomorphism, and let $V_p \in \operatorname{End}(J_0(N)_{\mathbb{F}_p})$ be its Verschiebung, so that

\begin{align*} F_p \circ V_p = V_p \circ F_p = [p]. \end{align*}

Let $T_p \in \operatorname{End}(J_0(N)_{\mathbb{F}_p})$ be the reduction modulo $p$ of the Hecke operator at $p$. Then

\begin{align*} T_p = F_p + V_p \end{align*}

in $\operatorname{End}(J_0(N)_{\mathbb{F}_p})$.

Equivalently, for every prime $\ell \ne p$, the induced endomorphism of the $\ell$-adic Tate module $T_\ell(J_0(N)_{\mathbb{F}_p})$ satisfies

\begin{align*} F_p^2 - T_p F_p + p = 0. \end{align*}

Consequently, if $A_f$ is the abelian quotient of $J_0(N)$ attached to a weight $2$ normalized Hecke eigenform $f$ with $T_p f = a_p f$, then the corresponding two-dimensional $\ell$-adic Galois representation has arithmetic Frobenius characteristic polynomial

\begin{align*} X^2 - a_p X + p. \end{align*}