\(\newcommand{\AA}{\mathcal{A}} \newcommand{\CH}{\mathcal{H}} \newcommand{\CL}{\mathcal{L}} \newcommand{\CO}{\mathcal{O}} \newcommand{\FF}{\mathbb{F}} \newcommand{\NN}{\mathbb{N}} \newcommand{\CC}{\mathbb{C}} \newcommand{\QQ}{\mathbb{Q}} \newcommand{\QQ}{\mathbb{Q}} \newcommand{\RR}{\mathbb{R}} \newcommand{\ZZ}{\mathbb{Z}}\) \(\DeclareMathOperator{\an}{an} \DeclareMathOperator{\char}{char} \DeclareMathOperator{\colim}{colim} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\Id}{Id} \DeclareMathOperator{\im}{im} \DeclareMathOperator{\Isom}{Isom} \DeclareMathOperator{\Gal}{Gal} \DeclareMathOperator{\GL}{GL} \DeclareMathOperator{\Ob}{Ob} \DeclareMathOperator{\Res}{Res} \DeclareMathOperator{\sh}{sh} \DeclareMathOperator{\SL}{SL} \DeclareMathOperator{\Spec}{Spec} \DeclareMathOperator{\Sym}{Sym}\)
Theorem 1.   Let \(f\) be an eigenform of weight \(k\). Then its Dirichlet series factors as

\[D_f(s) = \prod_{p \text{ prime}} (1-a_p p^{-s} + p^{k-1-2s})^{-1}.\]