Example ModFrm_Subspaces (H144E12)

We compute a basis of q-expansions for each of the above subspace of M₂(Γ₀(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)

The two projections sum to the original form.

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