.. index:: modular forms
*************
Modular forms
*************
One of 's computational specialities is (the very technical field
of) modular forms and can do a lot more than is even suggested in
this very brief introduction.
Cusp forms
==========
How do you compute the dimension of a space of cusp forms using Sage?
To compute the dimension of the space of cusp forms for Gamma use
the command ``dimension_cusp_forms``. Here is an example from
section "Modular forms" in the Tutorial:
::
sage: dimension_cusp_forms(Gamma0(11),2)
1
sage: dimension_cusp_forms(Gamma0(1),12)
1
sage: dimension_cusp_forms(Gamma1(389),2)
6112
Related commands: ``dimension_new__cusp_forms_gamma0`` (for
dimensions of newforms), ``dimension_modular_forms`` (for modular
forms), and ``dimension_eis`` (for Eisenstein series). The syntax is
similar - see the Reference Manual for examples.
In future versions of Sage, more related commands will be added.
.. index:: cosets of Gamma_0
Coset representatives
=====================
The explicit representation of fundamental domains of arithmetic
quotients :math:`H/\Gamma` can be determined from the cosets of
:math:`\Gamma` in :math:`SL_2(\ZZ)`. How are these cosets
computed in Sage?
Here is an example of computing the coset representatives of
:math:`SL_2(\ZZ)/\Gamma_0(11)`:
::
sage: G = Gamma0(11); G
Congruence Subgroup Gamma0(11)
sage: list(G.coset_reps())
[
[1 0] [ 0 -1] [1 0] [ 0 -1] [ 0 -1] [ 0 -1] [ 0 -1] [ 0 -1]
[0 1], [ 1 0], [1 1], [ 1 2], [ 1 3], [ 1 4], [ 1 5], [ 1 6],
[ 0 -1] [ 0 -1] [ 0 -1] [ 0 -1]
[ 1 7], [ 1 8], [ 1 9], [ 1 10]
]
.. index:: modular symbols, Hecke operators
Modular symbols and Hecke operators
===================================
Next we illustrate computation of Hecke operators on a space of
modular symbols of level 1 and weight 12.
::
sage: M = ModularSymbols(1,12)
sage: M.basis()
([X^8*Y^2,(0,0)], [X^9*Y,(0,0)], [X^10,(0,0)])
sage: t2 = M.T(2)
sage: f = t2.charpoly('x'); f
x^3 - 2001*x^2 - 97776*x - 1180224
sage: factor(f)
(x - 2049) * (x + 24)^2
sage: M.T(11).charpoly('x').factor()
(x - 285311670612) * (x - 534612)^2
Here ``t2`` represents the Hecke operator :math:`T_2` on the space
of Full Modular Symbols for :math:`\Gamma_0(1)` of weight
:math:`12` with sign :math:`0` and dimension :math:`3` over
:math:`\QQ`.
::
sage: M = ModularSymbols(Gamma1(6),3,sign=0)
sage: M
Modular Symbols space of dimension 4 for Gamma_1(6) of weight 3 with sign 0
and over Rational Field
sage: M.basis()
([X,(0,5)], [X,(3,2)], [X,(4,5)], [X,(5,4)])
sage: M._compute_hecke_matrix_prime(2).charpoly()
x^4 - 17*x^2 + 16
sage: M.integral_structure()
Free module of degree 4 and rank 4 over Integer Ring
Echelon basis matrix:
[1 0 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
See the section on modular forms in the Tutorial or the Reference
Manual for more examples.
Genus formulas
==============
Sage can compute the genus of :math:`X_0(N)`, :math:`X_1(N)`,
and related curves. Here are some examples of the syntax:
::
sage: dimension_cusp_forms(Gamma0(22))
2
sage: dimension_cusp_forms(Gamma0(30))
3
sage: dimension_cusp_forms(Gamma1(30))
9
See the code for computing dimensions of spaces of modular forms
(in ``sage/modular/dims.py``) or the paper by OesterlĂ© and Cohen {CO}
for some details.