### 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 |
---|---|

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 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

**Fourier expansions with modular form coefficients.** / ElGuindy, Ahmad.

Research output: Contribution to journal › Article

*International Journal of Number Theory*, vol. 5, no. 8, pp. 1433-1446. https://doi.org/10.1142/S1793042109002717

}

TY - JOUR

T1 - Fourier expansions with modular form coefficients

AU - ElGuindy, Ahmad

PY - 2009/12

Y1 - 2009/12

N2 - 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 X0(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 X0(l) has genus zero).

AB - 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 X0(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 X0(l) has genus zero).

KW - Atkin-Lehner involution

KW - Coefficient duality

KW - Fourier expansions

KW - Hyperelliptic modular curves

KW - Meromorphic modular forms

UR - http://www.scopus.com/inward/record.url?scp=77149134946&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77149134946&partnerID=8YFLogxK

U2 - 10.1142/S1793042109002717

DO - 10.1142/S1793042109002717

M3 - Article

AN - SCOPUS:77149134946

VL - 5

SP - 1433

EP - 1446

JO - International Journal of Number Theory

JF - International Journal of Number Theory

SN - 1793-0421

IS - 8

ER -