Package sage :: Package modular :: Package modform :: Module modform
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Module sage.modular.modform.modform

Modular Forms


DESIGN NOTES:
Our implementation depends heavily on the fact that we have good
dimension formulas (see dims.py) for spaces of modular forms with
character, and new subspaces, so that we don't have to compute
q-expansions for the whole space in order to compute q-expansions /
elements / and dimensions of certain subspaces.  Also, the following
design is much simpler than the one I used in MAGMA because subspaces
don't have lots of complicated special labels.  Here's the actual
design.

The internal basis of q-expansions of modular forms for the ambient
space is defined as follows:

  First Block:   Cuspidal Subspace
  Second Block:  Eisenstein Subspace

      Cuspidal Subspace:    Block for each level M dividing N, from highest
                            level to lowest, and block for level M
                            contains the images at level N of the
                            newsubspace of level M (basis, then
                            basis(q^d), then basis(q^e), etc.)
                            
     Eisenstein Subspace: ???

Since we can compute dimensions of cuspidal subspaces quickly and
easily, it should be easy to locate any of the above blocks.  Hence,
e.g., to compute basis for new cuspidal subspace, just have to return
first n standard basis vector where n is the dimension.  However, we
can also create completely arbitrary subspaces as well.

The base field is the field generated by the character values.

Classes
EisensteinSeries  
ModularFormElement A modular form.
ModularForms An ambient space of modular forms.
ModularFormsSpace A generic space of modular forms.
ModularFormsSubspace A subspace of an ambient space of modular forms.
ModularFormsWithCharacter A space of modular forms with character.

Function Summary
  compute_eisenstein_params(character, k)
Compute and return a list of all parameters (chi,psi,t) that define the Eisenstein series with given character and weight k.

Variable Summary
int DEFAULT_PRECISION = 6                                                                     

Function Details

compute_eisenstein_params(character, k)

Compute and return a list of all parameters (chi,psi,t) that define the Eisenstein series with given character and weight k.

Only the parity of k is relevant.

If character is an integer N, then the parameters for Gamma1(N) are computed instead. Then the condition is that chi(-1)*psi(-1) == (-1)**k.

Variable Details

DEFAULT_PRECISION

Type:
int
Value:
6                                                                     

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