Then the action of $\mathbb{T}$ on $A_f$ factors through $\mathbb{T}/I_f$, and, under the induced identification of $\mathbb{T}/I_f$ with its image in $\mathcal O_f$, the operator $T_n$ acts on $A_f$ as multiplication by the scalar $a_n(f)\in \mathcal O_f$ for every $n\geq 1$.