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 Gamma0(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].