This short but very beautiful paper contains the first step towards a solution of the deep conjecture of Atkin and Serre that \(| \tau(p)| \gg_{\varepsilon} p^{(9/2)-\varepsilon}\) (for every positive \(\varepsilon\) and prime \(p)\) where \(\tau(p)\) is the \(p\)th Fourier coefficient of the unique normalized cusp form of weight \(12\) for the full modular group. The main result is that whenever \(\tau(n)\) is odd it satisfies \(| \tau(n)| \geq (\log n)^c\) with some effectively computable positive absolute constant \(c\). The proof is straight-forward to follow but appeals to very deep results like \textit{A. Baker}'s estimate for linear forms [Acta Arith. 21, 117--129 (1972; Zbl 0244.10031)], \textit{N. I. Feld'man}'s and \textit{A. Baker}'s estimates on the magnitude of integral solutions to Thue's equation [Izv. Akad. Nauk. SSSR, Ser. Mat. 35, 973--990 (1971; Zbl 0237.10018) and Acta Arith. 24, 33--36 (1973; Zbl 0261.10025), respectively], and \textit{V. G. Sprindzhuk}'s estimate for the magnitude of integral solutions of hyperelliptic equations [Acta Arith. 30, 95--108 (1976; Zbl 0335.10021)]. It is a tour-de-force, tying together simple inequalities and very deep results to produce a clean and effective result. The methods also apply to the coefficients of more general modular forms.