# Newforms on Gamma_0(l) of Haupttypus

### Contents

### A short introduction

Table 1: Range of the Modi data base | ||
---|---|---|

weight k | level l |
range of a_f(n) |

2 | l ≤ 1000 | n ≤ 100 |

4 | l ≤ 200 | n ≤ 100 |

6 | l ≤ 120 | n ≤ 100 |

8 | l ≤ 60 | n ≤ 100 |

10 | l ≤ 40 | n ≤ 100 |

12 | l ≤ 30 | n ≤ 100 |

Currently the Modi data base contains the Fourier coefficients
`a_f(n)`
of all newforms `f = sum_n a_f(n) q^n` of integral weight `k`
and Haupttypus
on `Gamma _{0}(l)` within the range of Table 1.
For the origin of the data see
[Cohen-Skoruppa-Zagier].

The Hecke algebra, i.e. the algebra generated over the rational
numbers by the Hecke operators `T(n)` ( all `n`)
acting on the space of newforms
splits over the rationals into a direct product of totally real number
fields. Modi can be queried for equations defining these fields, their
discriminant and their Galois group, and the action of the
Atkin-Lehner operators `W_p` on the subspaces of newforms
belonging to these fields. Galois groups and discriminants are computed
for equations of degree up to 11.

### Searching for newforms by weight and level

Galois groups are denoted by a 3-component vector `[n,s,k]`, where `n` is
the order of the group, `s` is its signature, i.e. `s = 1` or `-1`
accordingly as the group is a
subgroup of the alternating group `A_n` or not, and where `k`
is the numbering of the group among all
transitive subgroups of `S_n` as given in
[Butler-McKay].