### Abstract

In this paper, we give a classification of weights k such that there is a nonzero weakly holomorphic modular form f=∑a(n)q^{n} of weight k on Γ_{1}(N) having infinitely many congruences of the form a(ℓn+β)≡0(modℓ), where ℓ is a prime and β is an integer in (0, 1, ..., ℓ-1). These are similar to congruences for the partition function investigated by Ramanujan. Furthermore, we characterize linear combinations of Shimura theta functions with odd characters in terms of these congruences. As an application of our main theorem, we consider a generalization of Newman's conjecture for weakly holomorphic modular forms on Γ_{0}(N) with real Dirichlet characters. Finally we use these results to study arithmetic properties of colored partitions and generalized Frobenius partitions.

Original language | English |
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Pages (from-to) | 489-516 |

Number of pages | 28 |

Journal | Advances in Mathematics |

Volume | 294 |

DOIs | |

Publication status | Published - 2016 May 14 |

Externally published | Yes |

### Keywords

- Colored partitions
- Congruences for modular forms
- Generalized Frobenius partitions

### ASJC Scopus subject areas

- Mathematics(all)