Combinations of cranks modulo 15

In this paper, we express the generating function of a combination of cranks of partitions that belong to congruence classes 3, 4, 5, 6, and 7 modulo 15, in terms of the Eisenstein series. Our results rely on the observation that the generating function of the combination we consider is a weight 1 modular form on Γ 0 ( 225 ) with the non-trivial modulo 3 primitive Dirichlet character. We obtain the modularity result by analyzing the transformation properties of related Klein forms. The main results are proven using the fact that the corresponding space is generated by Eisenstein series.