The following functions should only be called on modular symbols spaces that are cuspidal. For q-expansions of Eisenstein series, use the modular forms functions instead (see the example below).
The q-expansion of one of the Galois-conjugate newforms associated to the irreducible cuspidal space M of modular symbols, computed to absolute precision prec (which defaults to the highest precision computed in previous calls to this intrinsic, or 8 if none have been computed). The coefficients of the q-expansion lie in a quotient of a polynomial extension of the base field of M. In most cases, it is necessary for M to have been defined using NewformDecomposition.
Al: MonStgElt Default: em "Newform"
The reduced row-echelon basis of q-expansions for the space of modular forms associated to M, where K is the base field of M. The absolute precision of the q-expansions is prec.The optional parameter Al can take the values "Newform" and "Universal". The default is "Newform", which computes a basis of q-expansions by finding a decomposition of M into subspaces corresponding to newforms, computing their q-expansions, and then taking all of their images under the degeneracy maps. If Al := "Universal" then the algorithm of Section 4.3 of [Mer94] is used. This latter algorithm does not require computing a newform decomposition of M, but requires computing the action of many more Hecke operators. Consequently, in practice, our implementation of Merel's algorithm is usually less efficient than our implementation of the newform algorithm.
Al: MonStgElt Default: em "Newform"
The reduced integral basis of q-expansions for the space of modular forms associated to the given space M of modular symbols (or the given sequence of spaces). The q-expansions are computed to absolute precision prec. The base field of M must be either the rationals or a cyclotomic field.
The sequence of Hecke eigenvalues [a2, a3, a5, a7, ..., ap] attached to the space of modular symbols M, where p is the largest prime less than or equal to prec. Let K be the base field of M. Then the aell either lie in K or an extension of K (which may be constructed either as a number field or as a quotient of K[x]). We assume that M corresponds to a single Galois-conjugacy class of newforms.
> M := CuspidalSubspace(ModularSymbols(23,2, +1));
> qExpansionBasis(M);
[
q - q^3 - q^4 - 2*q^6 + 2*q^7 + O(q^8),
q^2 - 2*q^3 - q^4 + 2*q^5 + q^6 + 2*q^7 + O(q^8)
]
> f := qEigenform(M,6); f;
q + a*q^2 + (-2*a - 1)*q^3 + (-a - 1)*q^4 + 2*a*q^5 + O(q^6)
> Parent(f);
Power series ring in q over Univariate Quotient Polynomial Algebra
in a over Rational Field with modulus a^2 + a - 1
> PowerSeries(M);
q + a*q^2 + (-2*a - 1)*q^3 + (-a - 1)*q^4 + 2*a*q^5 + (a - 2)*q^6 +
(2*a + 2)*q^7 + O(q^8)
> SystemOfEigenvalues(M, 7);
[
a,
-2*a - 1,
2*a,
2*a + 2
]
Next we compare an integral and rational basis of q-expansions
for S2(Γ0(65)), computed using modular symbols.
> S := CuspidalSubspace(ModularSymbols(65,2,+1));
> qExpansionBasis(S);
[
q + 1/3*q^6 + 1/3*q^7 + O(q^8),
q^2 - 1/3*q^6 + 2/3*q^7 + O(q^8),
q^3 - 4/3*q^6 + 2/3*q^7 + O(q^8),
q^4 - 1/3*q^6 + 5/3*q^7 + O(q^8),
q^5 + 5/3*q^6 + 2/3*q^7 + O(q^8)
]
> qIntegralBasis(S);
[
q + q^5 + 2*q^6 + q^7 + O(q^8),
q^2 + 2*q^5 + 3*q^6 + 2*q^7 + O(q^8),
q^3 + 2*q^5 + 2*q^6 + 2*q^7 + O(q^8),
q^4 + 2*q^5 + 3*q^6 + 3*q^7 + O(q^8),
3*q^5 + 5*q^6 + 2*q^7 + O(q^8)
]
If you're interested in q-expansions of
Eisenstein series, see the chapter on modular forms.
For example:
> E := EisensteinSubspace(ModularForms(65,2));
> Basis(E);
[
1 + O(q^8),
q + 3*q^2 + 4*q^3 + 7*q^4 + 12*q^6 + 8*q^7 + O(q^8),
q^5 + O(q^8)
]