Chapter 17: Balanced Modular Parameterizations

For prime levels 5 ≤ p ≤ 19, sets of theta quotients are constructed that generate graded rings of modular forms of positive integer weight for Γ₁(p). Action by Γ₀(p) is shown to cyclically permute the generators. This induces symmetric representations for modular forms. The generators are used to deduce representations for the number of t-core partitions of an integer as convolutions of L-functions. Coupled systems of differential equations of level p are constructed for each basis analogous to Ramanujan’s differential equations for Eisenstein series on the full modular group.