| Filename |
File type |
Description |
| etamake0.txt |
Magma script |
This script takes as input a weight k and a level N
and uses LattE,
a program for enumerating lattice points in polytopes, to count the number of
eta-quotients in Mk(Γ0(N)). The LattE
executable count must be in the same directory in which magma is run for
this script to work. This script is only effective when the level N
is fairly small, but high weights k are feasible.
|
| etamake-lattice.txt |
Magma script |
This script takes as input a weight k and a level N
and enumerates short vectors in the lattice of weakly holomorphic eta-quotients
of level N to find all the eta-quotients in Mk(Γ0(N)). It does not rely on LattE, and it is most effective
when the weight is reasonably small and the level N does not have
too many divisors.
|
| etamake9.txt |
Magma script |
This script takes as input a range of levels N and tries
to prove that the graded ring of modular forms is generated by
holomorphic eta-quotients of level N. It attempts to find
enough eta-quotients to
span M2(Γ0(N)). To do this it
uses a combination of the short vectors method, and a random walk in
the lattice of weakly holomorphic eta-quotients to search for
holomorphic eta-quotients
in M2(Γ0(N)). If this random walk
is not successful, the program reverts to the method used in
etamake-lattice.txt. This script is effective for levels with lots of
small divisors. (Note: This script was written before the authors
realized then whenever N is composite
and Γ0(N), the graded ring of modular forms
is generated in weight 2. Hence, it includes a check that products of
weight 2 eta-quotients that were found span the weight 4 and weight 6
spaces.)
|
| etamake9.out |
Magma list |
This list is output by etamake9.txt. It gives a list of
the levels N with 1 ≤ N ≤ 524 for which
Γ0(N) has no elliptic points. For each level,
it indicates whether M2(Γ0(N)) is
spanned by eta-quotients, whether the graded ring of level N modular
forms is generated by eta-quotients, the dimension of M2(Γ0(N)), the dimension of the subspace spanned by eta-quotients,
and a lower bound for the number of eta-quotients in M2(Γ0(N)). (This lower bound is sharp if the space is not
|
| etamake9.data |
Text file |
This file lists output of the script etamake9.txt. For those levels N for which M2(Γ0(N)) is
spanned by eta-quotients, it lists eta-quotients that span the space.
|
| etamake10.txt |
Magma script |
This script attempts to prove that every modular form in Mk(Γ0(N)) is expressible as a linear combination of weakly
holomorphic eta-quotients. To do this, it tries to find eta-quotients that
span Mr+2(Γ0(N)), where r is
the weight of the form EN,N, the level N
eta-quotient with all of its zeroes at the cusp at infinity. It uses a similar
method to etamake9.txt.
|