Further Examples#
- Example: Orthogonal Rank4 Disc193 (ex-5e47d7)#
In this example we see how orthogonal modular forms of rank \(4\) correspond to Hilbert modular forms, by considering a rank \(4\) lattice with discriminant 193.
> Q := SymmetricMatrix([6,0,2,-4,-1,12,3,1,6,12]); > M := OrthogonalModularForms(Q); > Norm(Discriminant(Level(M))); 193 > time Dimension(M); 9 Time: 0.200 > T2 := HeckeOperator(M,2); > T2; [0 0 2 0 1 1 0 0 0] [0 3 0 1 1 0 0 0 0] [3 0 1 1 0 0 0 0 1] [0 3 2 2 2 2 1 0 0] [3 3 0 2 2 0 0 1 1] [3 0 0 2 0 2 2 0 1] [0 0 0 1 0 2 3 1 1] [0 0 0 0 1 0 1 5 1] [0 0 4 0 2 2 2 2 4] > D := Decomposition(M); > [Dimension(d) : d in D]; [ 1, 1, 7 ] > time fs := HeckeEigenforms(M); Time: 0.120 > #fs; 3 > [IsCuspidal(f) : f in fs]; [false, true, true]
So we have two irreducible Hecke modules in the cuspidal subspace, one of which is \(1\)-dimensional, and the other \(7\)-dimensional. Since the eigenforms correspond to orbit representatives under the Galois action, there is exactly one eigenform corresponding to each space, one with coefficients in \({\mathbb{Q}}\), and the other in a number field of degree \(7\).
We proceed to relate this last eigenforms to an elliptic cusp form \(g\) of level \(193\) and character \(\chi_{193}\), the Kronecker character of conductor \(193\), with LMFDB label 193.2.b.a. We first fix an embedding to identify the fields of definitions, and then verify that \(\lambda_p = a_p^2 + p(1-\chi(p))\), where \(\lambda_p\) are the eigenvalues of \(f\) and \(a_p\) are the eigenvalues of \(g\).
> f := fs[3]; > chi := KroneckerCharacter(193, Rationals()); > g := qEigenform(NewformDecomposition(CuspidalSubspace(ModularSymbols([chi],2))) > [1], 20); > L := Parent(Coefficient(g,1)); > K<alpha> := Parent(HeckeEigenvalue(f,2)); > rts := Roots(MinimalPolynomial(alpha), L); > h := hom<K -> L | rts[1][1]>; > for p in PrimesUpTo(20) do > ap := Coefficient(g,p); > lambda_p := HeckeEigenvalue(f,p); > assert ap^2 + p*(1-chi(p)) eq h(lambda_p); > end for;
We proceed to relate the remaining eigenform a to Hilbert modular form over the field \({\mathbb{Q}}(\sqrt{193})\), with LMFDB label 2.2.193-1.1.a. We verify that \(\lambda_p = a_{P} a_{\overline{P}}\) at split primes and \(\lambda_p = a_{P}\) at inert primes.
> f := fs[2]; > K := QuadraticField(193); > ZK := Integers(K); > g := Eigenforms(HilbertCuspForms(K, 1*ZK, [2,2]))[1]; > g_evs := [[HeckeEigenvalue(g, P) : P in PrimeIdealsOverPrime(K,p)] : > p in PrimesUpTo(10)]; > _<alpha> := Universe(g_evs[1]); > (2*alpha-1)^2; 17 > time &and[HeckeEigenvalue(f,p) eq &*[HeckeEigenvalue(g,P): > P in PrimeIdealsOverPrime(K,p)] : p in PrimesUpTo(20)]; true Time: 11.450
- Example: Unitary Rank3 Discm7 Example (ex-623abe)#
It is possible to use this package to compute unitary modular forms, and modular forms with nontrivial weight. Here we compute the spaces of unitary modular forms for the standard hermitian form over \({\mathbb{Q}}(\sqrt{-7})\), with varying weights.
Begin by considering trivial weight.
> K<sqrtm7> := QuadraticField(-7); > M0 := UnitaryModularForms(K,3); > Dimension(M0); 2 > time fs := HeckeEigenforms(M0); Time: 0.010 > fs; [ Eisenstein eigenform given in coordinates by (1 1), Cuspidal eigenform given in coordinates by ( 1 -1/7) ]
As observed in [Loeffler, 2008], and expanded upon in [Greenberg and Voight, 2014], the first system of Hecke eigenvalues is Eisenstein, and the second one is also a lift from \({\rm U}_{K/{\mathbb{Q}}}(1) \times {\rm U}_{K/{\mathbb{Q}}}(1) \times {\rm U}_{K/{\mathbb{Q}}}(1)\) of a character.
> time aPs, Ps := HeckeEigensystem(fs[1], 1 : Precision := 30); Time: 0.490 > aPs; [ 7, 7, 84, 630, 56, 133, 133, 553, 553, 871, 871 ] > aPs eq [IsSplit(P) select 1 + Norm(P) + Norm(P)^2 else IsInert(P) > select p + p^4 where _, p is IsPrincipal(P) else Norm(P) + Norm(P)^2 : P in Ps]; true > time bPs, Ps := HeckeEigensystem(fs[2], 1 : Precision := 30); Time: 0.000 > bPs; [ -1, -1, 20, 54, -8, 5, 5, 41, 41, -25, -25 ] > bPs eq [(IsSplit(P) select Trace(p^2) + Norm(P) else IsInert(P) > select 2*p^2+p-1 else p^2-1) where _, p is IsPrincipal(P) : P in Ps]; true
Notice that computing the eigenvalues \(b_P\) for the second form was instantaneous, as the Hecke operators were already computed and cached when computing the \(a_P\)’s.
We now consider unitary modular forms for the same space, but with weight \((3,0,-3)\). Note also that this is the representation denoted by \(V_{3,3,0}\) in [Loeffler, 2008] and [Greenberg and Voight, 2014].
> M33 := UnitaryModularForms(K,3,[3,0,-3],0); > time Dimension(M33); 4 Time: 1.000 > time D := Decomposition(M33); Time: 0.900 > [Dimension(d) : d in D]; [ 2, 2 ]
As observed in [Loeffler, 2008], we see that the action of the Hecke operators is not diagonalizable over \(K\), and the space \(M(3,0,-3) = W_1 \oplus W_2\) decomposes as a direct sum of two \(2\)-dimensional, Hecke-stable subspaces not diagonalizable over \(K\). One of these spaces arises as the lift from \({\rm U}_{K/{\mathbb{Q}}}(1) \times {\rm U}_{K/{\mathbb{Q}}}(2)\) involving an elliptic modular form in \(S_9(7, \chi_{-7})\), with LMFDB label 7.9.b.b, while the other is non-endoscopic. We proceed to verify the endoscopic space, corresponding to the elliptic modular form.
> fs := HeckeEigenforms(M33); > E1 := BaseField(fs[1]); > p_E<x> := DefiningPolynomial(E1); > E1; Number Field with defining polynomial x^2 - 6*x + 49/8 over K > IsIsomorphic(E1, ext<K | x^2 - 46>); true Mapping from: FldNum: E to Number Field with defining polynomial x^2 - 46 over K > SetVerbose("AlgebraicModularForms", 1); > time aPs33, Ps33 := HeckeEigensystem(fs[1], 1 : Precision := 30); Time: 99.240 > mfs := Newforms(CuspForms(KroneckerCharacter(-7, CyclotomicField(7)), 9)); > g := mfs[2][1]; > evs := [Norm(P) + Norm(P)^(-3)*Coefficient(g, Norm(P)) +(IsSplit(P) > select 0 else p-1 where _,p is IsPrincipal(P meet Integers())) : > P in Ps33 | not IsRamified(P)]; > E1 := AbsoluteField(Universe(aPs33)); > Eg := Universe(evs); > L := CompositeFields(E1, Eg)[1]; > aPs33_L := [L!aPs33[i] : i in [1..#aPs33] | not IsRamified(Ps33[i])]; > zeta := PrimitiveElement(E1); > roots := [x[1] : x in Roots(MinimalPolynomial(zeta), L)]; > embs := [hom<E1 -> L | r> : r in roots]; > exists(emb){emb : emb in embs | [emb(y) : y in aPs33_L] eq evs}; true
We are left with the non-endoscopic form. We check that the Hecke eigenvalue at the prime \(P_2\) above \(2\) is \({-7\pm\sqrt{-259}}\over{8}\). This shows that the Satake polynomial is irreducible, hence the form is not a lift, and the corresponding \(\ell\)-adic Galois representation is irreducible of dimension \(3\).
> E2 := BaseRing(fs[2]); > L<sqrtm259> := ext<K | x^2 + 259>; > is_isom, isom := IsIsomorphic(E2, L); > is_isom; true > isom(HeckeEigenvalue(fs[2], Ps[1])); -1/8*sqrtm259 - 7/8
Due to the large weight, the representation space is large (of dimension \(64\)), and computations of the Hecke eigenvalues are slow. However, we could compute their values modulo primes quite quickly by using representations over finite fields.
> M33_7 := UnitaryModularForms(K,3,[3,0,-3],7); > time D_7 := Decomposition(M33_7); Time: 3.250 > [Dimension(d) : d in D_7]; [ 1, 3 ] > fs_7 := HeckeEigenforms(M33_7); > time bPs_7, Ps_7 := HeckeEigensystem(fs_7[1], 1 : Precision := 30); Time: 14.770 > bPs_7; [ 6, 6, 6, 5, 5, 5, 6, 6, 3, 3 ] > ZK := Integers(K); > _, red := ResidueClassField(sqrtm7*ZK); > bPs_7 eq [red(bPs[i]) : i in [1..#bPs] | Ps[i] in Ps_7]; true > time aPs_7, Ps_7 := HeckeEigensystem(fs_7[2], 1 : Precision := 30); Time: 22.260 > aPs_7; [ 0, 0, 0, 0, 0, 0, 0, 0, 3, 3 ] > aPs_7 eq [red(aPs[i]) : i in [1..#aPs] | Ps[i] in Ps_7]; true
Therefore we obtain an explicit congruence modulo \(7\) between a nonendoscopic form in weight \((6,3,0)\) and an Eisenstein form in trivial weight. Thus, the mod \(7\) Galois representation associated to the system \(b_p\) is reducible.