Theorem 1.27 a) in Diamond, Darmon, Taylor, "Fermat's Last Theorem"

Question

I am looking for a reference for part a) of Theorem 1.27 here regarding the proof of the growth of coefficients of cusp forms. Theorem 1.24 gives a very sketchy argument of a less specific fact, but I lack the insight to see how to convert the explanation given there into Theorem 1.27 a). The only explanation I have been able to find is that this follows from the Weil Conjectures plus Modularity, but since the Diamond Darmon Taylor notes seek to establish a special case of Modularity, this explanation is certainly not suitable.