Design notes#

The implementation depends on the fact that we have dimension formulas (see 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 submodules don’t have lots of complicated special labels. A modular forms module can consist of the span of any elements; they need not be Hecke equivariant or anything else.

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. The 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:   characters, etc.

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 ring is the ring generated by the character values (or bigger). In MAGMA the base was always \(\ZZ\), which is confusing.