On a modular curve of arbitrary congruence level, we introduce the notion of a "Makdisi symbol", a device that simultaneously gives a moduli-friendly coordinate system while also having an elegant Hecke theory. Concretely, these are cuspidal projections of products of two weight 1 Eisenstein series, and their study was originally pioneered by the work of Kamal Khuri-Makdisi. We show that a certain subclass of symbols, the "invertible Makdisi symbols", yield precisely the eigenforms of rank zero; combining this with the moduli interpretation, we obtain a systematic and relatively efficient algorithm to determine the rational points on a modular curve, so long as the curve has a rank zero eigenform. The algorithm is $p$-adic in nature, based on the method of Chabauty-Coleman, and does not require finding any equations for the curve.