The \(p\)-adic eigencurve has been introduced by R. Coleman and B. Mazur in [Lond. Math. Soc. Lect. Note Ser. 254, 1–113 (1998; Zbl 0932.11030)], and presented open questions concerning it. Then, K. Buzzard and L. J. P. Kilford [Compos. Math. 141, No. 3, 605–619 (2005; Zbl 1187.11020)] have studied \(2\)-adic eigencurve and shown, for example, that the slopes of the overconvergent modular forms of weight \(k\) (in a limited region) are \(\{0, t, 2t, 3t, \ldots, \}\) with multiplicity one, where \(t\) is determined by \(k\). In the paper under review, the author follows the method of Buzzard and Kilford, and shows similar results on the \(3\)-adic eigencurve.