Distinguishing eigenforms modulo a prime ideal

Abstract

Consider the Fourier expansions of two elements of a given space of modular forms. How many leading coefficients must agree in order to guarantee that the two expansions are the same? Sturm gave an upper bound for modular forms of a given weight and level. This was adapted by Ram Murty, Kohnen and Ghitza to the case of two eigenforms of the same level but having potentially different weights. We consider their expansions modulo a prime ideal, presenting a new bound. In the process of analysing this bound, we generalise a result of Bach and Sorenson, who provide a practical upper bound for the least prime in an arithmetic progression.