Consider an ambient space M1 of modular symbols of level N1, and suppose M2 is an ambient space of modular symbols of level a multiple N2 of N1 whose weight equals the weight of M1 and whose character is induced by the character of M1. Then for each divisor d of N2/N1 there are natural maps αd:M1 -> M2 and βd:M2 -> M1 such that βd αd is multiplication by dk - 2.[Γ0(N1) : Γ0(N2)], where k is the common weight of M1 and M2. On cuspidal parts, the map βd is dual to the map f(q) -> f(qd) on modular forms. Use the function DegeneracyMap to compute the maps αd and βd.
Given a space M of modular symbols and a positive integer N that is a multiple of the level of M, the images of M under the degeneracy maps generate a modular symbols space of level N. The constructor ModularSymbols(M,N) computes this space.
Let M be a space of modular symbols of level N, and let N' be a multiple of N. The subspace ∑_(d | (N'/N)) αd(M) ⊂Mmk(N', ε) is stable under the Hecke operators. Here is how to create this subspace using Magma:
> M := ModularSymbols(11,2); M; Full modular symbols space for Gamma_0(11) of weight 2 and dimension 3 over Rational Field > M33 := ModularSymbols(M,33); M33; Modular symbols space for Gamma_0(33) of weight 2 and dimension 6 over Rational Field
The degeneracy map M1 -> M2 of spaces of modular symbols associated to d. Let Ni be the level of Mi for i=1, 2. Suppose that d is a divisor of either the numerator or denominator of the rational number N1/N2, written in reduced form. If N1 | N2, then this intrinsic returns αd:M1 -> M2, or if N2 | N1, then this intrinsic returns βd:M1 -> M2. It is an error if neither divisibility holds.
Given spaces of modular symbols M1 and M2 and an integer d, return the matrix of DegeneracyMap(M1,M2,d) with respect to Basis(M1) and Basis(M2). Both IsAmbient(M1) and IsAmbient(M2) must be true.
The modular symbols space of level N' associated to M. Let N be the level of M. If N | N', then this intrinsic returns the modular symbols space ∑_(d | (N'/()N)) αd(M). If N' | N, then this intrinsic returns the modular symbols space ∑_(d | (N/()N')) βd(M). In this latter case, if Conductor(DirichletCharacter(M)) does not divide N', then the 0 space is returned.
The modular symbols subspace of M1 associated to M2. Let N1 be the level of M1. If ModularSymbols(M2,N1) is defined, let M3 be this modular symbols space, otherwise terminate with an error. If M3 is contained in M1, return M3, otherwise terminate with an error.
> M15 := ModularSymbols(15);
> M30 := ModularSymbols(30);
> alp_2 := DegeneracyMap(M15,M30,2);
> alp_2(M15.1);
2*{oo, 0} + -1*{-1/28, 0} + -1*{-1/2, -7/15}
> beta_2 := DegeneracyMap(M30,M15,2);
> beta_2(alp_2(M15.1));
3*{oo, 0}
> M15.1;
{oo, 0}
We can consider the space generated by the image of a space of modular symbols of level 11 in spaces of higher level.
> X11 := ModularSymbols("11k2A");
> qEigenform(X11,6);
q - 2*q^2 - q^3 + 2*q^4 + q^5 + O(q^6)
> ModularSymbols(X11,33);
Modular symbols space for Gamma_0(33) of weight 2 and dimension 4 over
Rational Field
> X33 := ModularSymbols(X11,33);
> qExpansionBasis(X33,6);
[
q - 2*q^2 + 2*q^4 + q^5 + O(q^6),
q^3 + O(q^6)
]
> Factorization(CharacteristicPolynomial(HeckeOperator(X33,3)));
[
<x^2 + x + 3, 2>
]
> ModularDegree(X33);
3
We can also construct the space generated by the images
of X11 at higher level using the !! operator.
> M44 := ModularSymbols(44,2); > A := M44!!X11; A; Modular symbols space for Gamma_0(44) of weight 2 and dimension 6 over Rational Field > X11!!A; // back to the original space Modular symbols space for Gamma_0(11) of weight 2 and dimension 2 over Rational Field