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Creation Functions

Subsections

Ambient Spaces

The following are used for creating spaces of modular forms. For information on Dirichlet characters, see Section Dirichlet Characters.

ModularForms(N) : RngIntElt -> ModFrm
The space M20(N), Z) of modular forms on Γ0(N) of weight 2. See the documentation for ModularForms(N,k) below, with k=2.
ModularForms(N, k) : RngIntElt, RngIntElt -> ModFrm
The space Mk0(N), Z) of weight k modular forms on Γ0(N) over Z.
ModularForms(eps, k) : GrpDrchElt, RngIntElt -> ModFrm
Given a Dirichlet character eps and an integer k, this returns a space of modular forms over the integers, of weight k, which under base extension becomes equal to the direct sum of the spaces Mk1(N), eps1) of weight k and nebentypus character eps1, where eps1 runs over all Galois conjugates eps.
ModularForms(chars, k) : [GrpDrchElt], RngIntElt -> ModFrm
The space of modular forms of weight k over the integers, formed as the direct sum of spaces ModularForms(eps,k), summing over all eps in the given sequence chars of Dirichlet characters.
ModularForms(G) : -> ModFrm
This is the same as ModularForms(G,2) (see below).
ModularForms(G, k) : -> ModFrm
The space Mk(G, Z), where G is a congruence subgroup. The groups Γ0(N) and Γ1(N) are currently supported, and can be created using the commands Gamma0(N) and Gamma1(N), respectively.
CuspForms(x) : Any -> ModFrm
CuspForms(x,y) : Any, Any -> ModFrm
These commands are a shortcut, and return the CuspidalSubspace of the corresponding full space of modular forms.

Example ModFrm_Creation-Space (H132E3)

In this example, we illustrate each of the above constructors in turn. First we create M20(65)).

> M := ModularForms(65); M;
Space of modular forms on Gamma_0(65) of weight 2 and dimension 8 over
Integer Ring.
> Dimension(M);
8
> Basis(CuspidalSubspace(M));
[
    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)
]
Next we create M40(8)).

> M := ModularForms(8,4); M;
Space of modular forms on Gamma_0(8) of weight 4 and dimension 5 over 
Integer Ring.
> Dimension(M);
5
> Basis(CuspidalSubspace(M));
[
    q - 4*q^3 - 2*q^5 + 24*q^7 + O(q^8)
]

Now we create the space M3(N, varepsilon), where varepsilon is a character of level 20, conductor 5 and order 4.

> G := DirichletGroup(20,CyclotomicField(EulerPhi(20)));
> chars := Elements(G); #chars;
8
> [Conductor(eps) : eps in chars];
[ 1, 4, 5, 20, 5, 20, 5, 20 ]
> eps := chars[3];
> IsEven(eps);          
false
> M := ModularForms([eps],3); M;
Space of modular forms on Gamma_1(20) with character all conjugates of
[$.2], weight 3, and dimension 12 over Integer Ring.
> Dimension(EisensteinSubspace(M));
6
> Dimension(CuspidalSubspace(M));
6
Next we create the direct sum of the spaces Mk(20, varepsilon) as varepsilon varies over the four mod 20 characters of order at most 2, for k=2 and 3.

> G := DirichletGroup(20, RationalField());   // (Z/20Z)^* --> Q^*
> chars := Elements(G); #chars;
4
> M := ModularForms(chars,2); M;
Space of modular forms on Gamma_1(20) with characters all
conjugates of [1, $.1, $.2, $.1*$.2], weight 2, and dimension 12
over Integer Ring.
> M := ModularForms(chars,3); M;
Space of modular forms on Gamma_1(20) with characters all
conjugates of [1, $.1, $.2, $.1*$.2], weight 3, and dimension 16
over Integer Ring.
Now we create the spaces Mk1(20)) for k=2, 3.

> ModularForms(Gamma1(20)); 
Space of modular forms on Gamma_1(20) of weight 2 and dimension 22 
over Integer Ring.
> ModularForms(Gamma1(20),3); 
Space of modular forms on Gamma_1(20) of weight 3 and dimension 34 
over Integer Ring.
We can also create the subspace of cuspforms directly:

> CuspForms(Gamma1(20)); 
Space of modular forms on Gamma_1(20) of weight 2 and dimension 3
over Integer Ring.
> CuspForms(Gamma1(20),3);
Space of modular forms on Gamma_1(20) of weight 3 and dimension 14 
over Integer Ring.

Half-integral Weight Forms

Spaces of modular forms of half-integral weight can now be constructed. For these spaces, CuspidalSubspace and qExpansionBasis are available, as well as basic functionality such as element arithmetic. More functionality will be added in future releases.

The algorithm for determining the q-expansion basis involves computing those of related integral-weight spaces (of weight either one half smaller or one half larger, and appropriate level and character).

HalfIntegralWeightForms(N, w) : RngIntElt, FldRatElt -> ModFrm
The space of half-integral weight forms on Gamma0(N) and weight w. Here N should be a multiple of 4 and w a positive element of Z + 1/2.
HalfIntegralWeightForms(chi, w) : GrpDrchElt, FldRatElt -> ModFrm
The space of half-integral weight forms on Gamma1(N) with character chi and weight w. The modulus of chi should be a multiple of 4, and w a positive element of Z + 1/2.
HalfIntegralWeightForms(G, w) : GrpPSL2, FldRatElt -> ModFrm
The space of half-integral weight forms on the congruence subgroup G and weight w. Here G must be contained in Gamma0(4), and w is a positive element of Z + 1/2.

Base Extension

If M is a space of modular forms created using one of the constructors in Section Ambient Spaces, then the base ring of M is Z. Thus we can base extend M to any ring R. The examples below illustrate some simple applications of BaseExtend.

BaseExtend(M, R) : ModFrm, Rng -> ModFrm, Map
The base extension of the space M of modular forms to the ring R and the induced map from M to BaseExtend(M,R). The only requirement on R is that there is a natural coercion map from the base ring of M to R. For example, when BaseRing(M) is the integers, any ring R is allowed.
BaseExtend(M, phi) : ModFrm, Map -> ModFrm, Map
The base extension of the space M of modular forms to the ring R using the map φ: BaseRing(M) -> R, and the induced map from M to BaseExtend(M,R)

Example ModFrm_BaseExtend (H132E4)

We first illustrate an Eisenstein series in M12(1) that is congruent to 1 modulo 3.

> M<q> := EisensteinSubspace(ModularForms(1,12));
> E12 := M.1; E12 + O(q^4);
691 + 65520*q + 134250480*q^2 + 11606736960*q^3 + O(q^4)
> M3<q3> := BaseExtend(M,GF(3));
> Dimension(M3);
1
> M3.1+O(q3^20);
1 + O(q3^20)
This congruence can be proved by noting that the coefficient of qn in the q-expansion of E12/65520, for any n≥1, is an eigenvalue of a Hecke operator, hence an integer, and that 65520 is divisible by 3. Because E12 is defined over Z the command "E12Q/65520" would result in an error, so we first base extend to Q.

> MQ, phi := BaseExtend(M,RationalField());
> E12Q := phi(E12);   
> E12Q/65520;
691/65520 + q + 2049*q^2 + 177148*q^3 + 4196353*q^4 + 48828126*q^5 + 
362976252*q^6 + 1977326744*q^7 + O(q^8)

It is possible to base extend to almost any silly commutative ring.

> M := ModularForms(11,2);
> R := PolynomialRing(GF(17),3);
> MR<q> := BaseExtend(M,R); MR;
Space of modular forms on Gamma_0(11) of weight 2 and dimension 2 over
Polynomial ring of rank 3 over GF(17)
Lexicographical Order
Variables: $.1, $.2, $.3.
> f := MR.1;  f + O(q^5);
1 + 12*q^2 + 12*q^3 + 12*q^4 + O(q^5)
> f*(R.1+3*R.2) + O(q^4);
$.1 + 3*$.2 + (12*$.1 + 2*$.2)*q^2 + (12*$.1 + 2*$.2)*q^3 + O(q^4)

Elements

M . i : ModFrm, RngIntElt -> ModFrmElt
The ith basis vector of the space of modular forms M.
M ! f : ModFrm, . -> ModFrmElt
The coercion of f into the space of modular forms M. Here f can be a modular form, a power series with absolute precision, or something that can be coerced into RSpace(M).
ModularForm(E) : CrvEll -> ModFrm
The modular form associated to the elliptic curve E over Q. (See Section Elliptic Curves.)

Example ModFrm_CreationElements (H132E5)

> M := ModularForms(Gamma0(11),2);
> M.1;
1 + 12*q^2 + 12*q^3 + 12*q^4 + 12*q^5 + 24*q^6 + 24*q^7 + O(q^8)
> M.2;
q - 2*q^2 - q^3 + 2*q^4 + q^5 + 2*q^6 - 2*q^7 + O(q^8)
> R<q> := PowerSeriesRing(Integers());
> f := M!(1 + q + 10*q^2 + O(q^3));
> f;
1 + q + 10*q^2 + 11*q^3 + 14*q^4 + 13*q^5 + 26*q^6 + 22*q^7 + O(q^8)
> Eltseq(f);
[ 1, 1 ]
Eltseq gives f as a linear combination of M.1 and M.2. Next we coerce f into M20(22)).

> M22 := ModularForms(Gamma0(22),2);
> g := M22!f; g;
1 + q + 10*q^2 + 11*q^3 + 14*q^4 + 13*q^5 + 26*q^6 + 22*q^7 + O(q^8)
> Eltseq(g);
[ 1, 1, 10, 11, 14 ]
The elliptic curve E below defines an element of M20(11)).

> E := EllipticCurve([ 0, -1, 1, -10, -20 ]);
> Conductor(E);
11
> f := ModularForm(E);
> f;
q - 2*q^2 - q^3 + 2*q^4 + q^5 + 2*q^6 - 2*q^7 + O(q^8)
> f + M.1;
1 + q + 10*q^2 + 11*q^3 + 14*q^4 + 13*q^5 + 26*q^6 + 22*q^7 + O(q^8)
Note, however, that the ambient space of the parent of f is not equal to M, despite the fact that both are isomorphic to M20(11)). This is because they were created independently. The above operations are defined because there is a canonical way to coerce f into M using its q-expansion.

> f in M;
false
> AmbientSpace(Parent(f)) eq M;
false

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Version: V2.19 of Wed Apr 24 15:09:57 EST 2013