### Abstract

In this paper, we study the Fourier expansion where the coefficients are given as the evaluation of a sequence of modular forms at a fixed point in the upper half-plane. We show that for prime levels l for which the modular curve X_{0}(l) is hyperelliptic (with hyperelliptic involution of the AtkinLehner type) then one can choose a sequence of weight k (any even integer) forms so that the resulting Fourier expansion is itself a meromorphic modular form of weight 2-k. These sequences have many interesting properties, for instance, the sequence of their first nonzero next-to-leading coefficient is equal to the terms in the Fourier expansion of a certain weight 2-k form. The results in the paper generalizes earlier work by Asai, Kaneko, and Ninomiya (for level one), and Ahlgren (for the cases where X_{0}(l) has genus zero).

Original language | English |
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Pages (from-to) | 1433-1446 |

Number of pages | 14 |

Journal | International Journal of Number Theory |

Volume | 5 |

Issue number | 8 |

DOIs | |

Publication status | Published - Dec 2009 |

Externally published | Yes |

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### Keywords

- Atkin-Lehner involution
- Coefficient duality
- Fourier expansions
- Hyperelliptic modular curves
- Meromorphic modular forms

### ASJC Scopus subject areas

- Algebra and Number Theory