Subspaces#

The following functions compute the cuspidal, Eisenstein, and new subspaces.

ZeroSubspace(M): ModFrm -> ModFrm#: The trivial subspace of the space of modular forms \(M\).

CuspidalSubspace(M): ModFrm -> ModFrm#: The subspace of forms \(f\) in \(M\) such that the constant term of the Fourier expansion of \(f\) at every cusp is \(0\).

EisensteinSubspace(M): ModFrm -> ModFrm#: The Eisenstein subspace of the space of modular forms \(M\).

EisensteinProjection(f): ModFrmElt -> ModFrmElt#

CuspidalProjection(f): ModFrmElt -> ModFrmElt#

The projection of a given modular form to the EisensteinSubspace or to the CuspidalSubspace. The sum of the two projections equals the original form (after coercion).

The base ring of the given form must contain the rationals.

NewSubspace(M): ModFrm -> ModFrm#: The new subspace of the space of modular forms \(M\).

DihedralSubspace(M): ModFrm -> ModFrm#: For a space \(M\) of weight \(1\) forms, this returns the subspace spanned by the cusp forms attached to dihedral Galois representations.

Example: Subspaces (ex-fd9fb5)#

We compute a basis of \(q\)-expansions for each of the above subspace of \(M_2(\Gamma_0(33))\).

> M := ModularForms(Gamma0(33),2); M;
Space of modular forms on Gamma_0(33) of weight 2 and dimension 6 over
Integer Ring.
> Basis(M);
[
    1 + O(q^8),
    q - q^5 + 2*q^7 + O(q^8),
    q^2 + 2*q^7 + O(q^8),
    q^3 + O(q^8),
    q^4 + q^5 + O(q^8),
    q^6 + O(q^8)
]
> Basis(CuspidalSubspace(M));
[
    q - q^5 - 2*q^6 + 2*q^7 + O(q^8),
    q^2 - q^4 - q^5 - q^6 + 2*q^7 + O(q^8),
    q^3 - 2*q^6 + O(q^8)
]
> Basis(EisensteinSubspace(M));
[
    1 + O(q^8),
    q + 3*q^2 + 7*q^4 + 6*q^5 + 8*q^7 + O(q^8),
    q^3 + 3*q^6 + O(q^8)
]
> Basis(NewSubspace(M));
[
    q + q^2 - q^3 - q^4 - 2*q^5 - q^6 + 4*q^7 + O(q^8)
]
> Basis(NewSubspace(EisensteinSubspace(M)));
[]
> Basis(NewSubspace(CuspidalSubspace(M)));
[
    q + q^2 - q^3 - q^4 - 2*q^5 - q^6 + 4*q^7 + O(q^8)
]
> ZeroSubspace(M);
Space of modular forms on Gamma_0(33) of weight 2 and dimension 0 over
Integer Ring.
> MQ := BaseChange(M, Rationals()); SetPrecision(MQ, 20);
> b := Basis(MQ); b[5];
q^4 + q^5 + 2*q^8 - q^9 + 2*q^10 + 2*q^11 - q^12 + 2*q^13 +
 2*q^14 - q^15 + 3*q^16 + 2*q^17 - 2*q^18 + 2*q^19 + O(q^20)
> CuspidalProjection(b[5]);
-1/10*q - 3/10*q^2 + 1/10*q^3 + 3/10*q^4 + 2/5*q^5 + 3/10*q^6 - 4/5*q^7
+ 1/2*q^8 - 3/10*q^9 + 1/5*q^10 - 1/10*q^11 - 3/10*q^12 + 3/5*q^13 -
2/5*q^14 - 2/5*q^15 - 1/10*q^16 + 1/5*q^17 + 1/10*q^18 + O(q^20)
> EisensteinProjection(b[5]);
1/10*q + 3/10*q^2 - 1/10*q^3 + 7/10*q^4 + 3/5*q^5 - 3/10*q^6 + 4/5*q^7
+ 3/2*q^8 - 7/10*q^9 + 9/5*q^10 + 21/10*q^11 - 7/10*q^12 + 7/5*q^13 +
12/5*q^14 - 3/5*q^15 + 31/10*q^16 + 9/5*q^17 - 21/10*q^18 + 2*q^19 +
O(q^20)
> MQ! $1 + MQ! $2;  // Add the previous two answers, inside MQ
q^4 + q^5 + 2*q^8 - q^9 + 2*q^10 + 2*q^11 - q^12 + 2*q^13 +
 2*q^14 - q^15 + 3*q^16 + 2*q^17 - 2*q^18 + 2*q^19 + O(q^20)

Run in calculator

The two projections sum to the original form.