### MAGMA package *FastBases*

#### Introduction

This Magma package has been designed for computing bases of the modular forms spaces M_k(Gamma_0(4),Q) for half-integral weights k. The basis modular forms are represented by their q-expansions up to a given precision. The main purpose is to be able to obtain a high precision.
The algorithms are described in the paper "Fast computation of half-integral weight modular forms" by Ilker Inam and Gabor Wiese.

#### Download

Download the complete code, as well as the documentation and an example as a tar-file (last update 31 August 2022).

#### Files

FastBases.spec | Package file. |

FB_General.m | Functions for various uses. |

FB_Kohnen_Basis.m | Functions for computing the Kohnen basis. |

FB_Rankin_Cohen.m | Functions for computing the Rankin-Cohen basis. |

FB_Standard_Basis.m | Functions for computing the standard basis. |

FB_standard_forms.m | Functions for computing some standard modular forms. |

FB-Example.m | An example illustrating the main functionality. |

User-Guide.txt | Short user guide.. |

#### Installation

It suffices to change to the directory containing the files of the package and to type in Magma:

AttachSpec("FastBases.spec");

#### Principal functions

FB_standard_basis(k,prec);

For any k half-integer or integer k, this function computes the q-expansions of the standard basis of the full space M_k(Gamma_0(4)) up to precision prec.

FB_Kohnen_basis(k,prec);

For a half-integer k = l+1/2 with l an integer at least 12, this function computes the q-expansions of the Kohnen basis of the plus space M_k^+(Gamma_0(4)) up to precision prec.

FB_RC_basis_plus(k,prec);

For a half-integer k = l+1/2 with l an even integer, this function computes the q-expansions of the Rankin-Cohen basis of the plus space M_k^+(Gamma_0(4)) up to precision prec.

FB_RC_basis_full(k,prec);

For a half-integer k = l+1/2 with l an odd integer, this function computes the q-expansions of the Rankin-Cohen basis of the full space M_k(Gamma_0(4)) up to precision prec.

FB_plus_subspace(C,k);

Given a half-integral or integral weight k and a basis C of q-expansions of weight k modular forms, this function computes a basis of the plus subspace spanned by the forms in C.

Last modification: 31 August 2022.