On the Miller basis for the space of cusp forms

Abstract

In this paper, we study the existence of the Miller basis for [Formula: see text], the space of cusp forms of weight [Formula: see text] and level [Formula: see text]. We seek an easy criterion to determine whether or not the space [Formula: see text] admits the Miller basis. In particular, we prove that if the level [Formula: see text] is sufficiently composite, then [Formula: see text] does not have the Miller basis. We also prove that there exists a constant [Formula: see text] such that [Formula: see text] admits the Miller basis if and only if [Formula: see text] admits the Miller basis when [Formula: see text].