Creation Functions#
Ambient Spaces#
The functions in this section are used to create spaces of modular forms. Spaces of half-integral weight can also be created (see the next section).
For information on Dirichlet characters, see Section Dirichlet Characters.
- ModularForms(N): RngIntElt -> ModFrm#
The space \(M_2(\Gamma_0(N),{\mathbb{Z}})\) of modular forms on \(\Gamma_0(N)\) of weight \(2\). See the documentation for
ModularForms(N,k)below, with \(k=2\).
- ModularForms(N, k): RngIntElt, RngIntElt -> ModFrm#
The space \(M_k(\Gamma_0(N),{\mathbb{Z}})\) of weight \(k\) modular forms on \(\Gamma_0(N)\) over \({\mathbb{Z}}\).
- 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 \(M_k(\Gamma_1(N),eps1)\) 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, k): GrpPSL2, RngIntElt -> ModFrm#
- ModularForms(G): GrpPSL2 -> ModFrm#
The space \(M_k(G,{\mathbb{Z}})\), where \(G\) is a congruence subgroup. The groups \(\Gamma_0(N)\) and \(\Gamma_1(N)\) are currently supported, and can be created using the commands
Gamma0(N)andGamma1(N), respectively. When not specified, \(k = 2\).
- CuspForms(x): Any -> ModFrm#
- CuspForms(x, y): Any, Any -> ModFrm#
These commands are a shortcut, and return the
CuspidalSubspaceof the corresponding full space of modular forms.
- Example: Creation Space (ex-902ab1)#
In this example, we illustrate each of the above constructors in turn. First we create \(M_2(\Gamma_0(65))\).
> 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 \(M_4(\Gamma_0(8))\).
> 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 \(M_3(N,\varepsilon)\), 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 ] > exists(eps){eps : eps in chars | Conductor(eps) eq 5 and IsOdd(eps)}; true > Order(eps); 4 > M := ModularForms(eps, 3); M; Space of modular forms on Gamma_1(20) with character all conjugates of [$.2], weight 3, and dimension 18 over Integer Ring. > Dimension(EisensteinSubspace(M)); 12 > Dimension(CuspidalSubspace(M)); 6
Next we create the direct sum of the spaces \(M_k(20,\varepsilon)\) 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 \(M_k(\Gamma_1(20))\) 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 also 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 \({\mathbb{Z}}+ 1/2\).
- HalfIntegralWeightForms(chi, w): GrpDrchElt, FldRatElt -> ModFrm#
The space of half-integral weight forms on
Gamma1(N)with characterchiand weight \(w\). The modulus ofchishould be a multiple of \(4\), and \(w\) a positive element of \({\mathbb{Z}}+ 1/2\).
- 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 \({\mathbb{Z}}+ 1/2\).
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 \({\mathbb{Z}}\). 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, whenBaseRing(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 \(\phi \colon\)
BaseRing(M)\(\rightarrow R\), and the induced map from \(M\) toBaseExtend(M,R)
- Example: Base Extend (ex-c5a11c)#
We first illustrate an Eisenstein series in \(M_{12}(1)\) 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 \(q^n\) in the \(q\)-expansion of \(E_{12}/65520\), for any \(n\geq 1\), is an eigenvalue of a Hecke operator, hence an integer, and that \(65520\) is divisible by \(3\). Because \(E_{12}\) is defined over \({\mathbb{Z}}\) the command
"E12Q/65520"would result in an error, so we first base extend to \({\mathbb{Q}}\).> 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 \(i\)th 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 \({\mathbb{Q}}\). (See Section Elliptic Curves.)
- Example: Creation Elements (ex-e76151)#
> 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 ]
Eltseqgives \(f\) as a linear combination ofM.1andM.2. Next we coerce \(f\) into \(M_2(\Gamma_0(22))\).> 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 \(E\) defines an element of \(M_2(\Gamma_0(11))\).
> 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)
A new copy of \(S_2(\Gamma_0(11))\) was created as the space containing \(f\)
> Sf := Parent(f); > Mf := AmbientSpace(Parent(f)); > Sf; Mf; Space of modular forms on Gamma_0(11) of weight 2 and dimension 1 over Integer Ring Space of modular forms on Gamma_0(11) of weight 2 and dimension 2 over Integer Ring > IsIdentical(M, Mf); false > M eq Mf; true > f in Sf, f in Mf, f in M; true false false
There is a canonical way to coerce \(f\) into \(M\) using its \(q\)-expansion.
> IsCoercible(M, f); true q - 2*q^2 - q^3 + 2*q^4 + q^5 + 2*q^6 - 2*q^7 - 2*q^9 - 2*q^10 + q^11 + O(q^12)
This coercion is applied automatically for some operations.
> f + M.1; 1 + q + 10*q^2 + 11*q^3 + 14*q^4 + 13*q^5 + 26*q^6 + 22*q^7 + 36*q^8 + 34*q^9 + 46*q^10 + q^11 + O(q^12)