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Subspaces






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



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 M.  



NewSubspace(M) : ModFrm-> ModFrm



The new subspace of M.  



ZeroSubspace(M) : ModFrm  -> ModFrm



The trivial subspace of M.





Example ModForm_Subspaces (H90E11)

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.





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