> S := CuspidalSubspace(ModularSymbols(Gamma1(7), 5, 1)); > newforms := NewformDecomposition(S); > Eigenform(newforms[1], 15); q + q^2 - 15*q^4 + 49*q^7 - 31*q^8 + 81*q^9 - 206*q^11 + 49*q^14 + O(q^15) > pi := LocalComponent(newforms[1], 7); > pi; Ramified Principal Series Representation of GL(2,Q_7) > chi := CentralCharacter(pi); > Conductor(chi); 7 > parameters := PrincipalSeriesParameters(pi);These are Dirichlet characters on Z/7Z (the trivial character and the character of order 2):
> Conductor(parameters[1]), Order(parameters[1]); 1 1 > Conductor(parameters[2]), Order(parameters[2]); 7 2The principal series representation πis the induction up to GL2(Q7) of a character of the Borel subgroup inflated from a character of the diagonal group Q7 x x Q7 x . The restriction of this character to Z7 x x Z7 x gives the pair of Dirichlet characters above. We now compute the Galois representation.
> G, map, F, rho := WeilRepresentation(pi); > F; Totally ramified extension defined by the polynomial x^6 + 7*x^5 + 21*x^4 + 35*x^3 + 35*x^2 + 21*x + 7 over 7-adic ring mod 7^10 > IsAbelian(G); trueThe Weil representation is simply the sum of the two characters above, considered as characters of the Galois group of Q7 via local class field theory.
> IsIrreducible(rho); false GModule of dimension 1 over Cyclotomic Field of order 6 and degree 2 GModule of dimension 1 over Cyclotomic Field of order 6 and degree 2
> S := CuspidalSubspace(ModularSymbols(Gamma0(121), 2, 1)); > newforms := NewformDecomposition(S); > newforms; [ Modular symbols space for Gamma_0(121) of weight 2 and dimension 1 over Q, Modular symbols space for Gamma_0(121) of weight 2 and dimension 1 over Q, Modular symbols space for Gamma_0(121) of weight 2 and dimension 1 over Q, Modular symbols space for Gamma_0(121) of weight 2 and dimension 1 over Q, Modular symbols space for Gamma_0(121) of weight 2 and dimension 2 over Q ] > Eigenform(newforms[2], 11); q + q^2 + 2*q^3 - q^4 + q^5 + 2*q^6 - 2*q^7 - 3*q^8 + q^9 + q^10 + O(q^11) > pi := LocalComponent(newforms[2], 11); > pi; Supercuspidal Representation of GL(2,Q_11)This means the representation of the Weil group associated to pi is irreducible.
> Conductor(pi); 121 > W := CuspidalInducingDatum(pi); > W; GModule W of dimension 10 over Rational FieldW is a module over a group which is a quotient of GL2(Z11), namely GL2(Z/(11Z)). The representation πis induced from some extension of W to the open subgroup Q11 x GL2(Z11).
> Group(W); MatrixGroup(2, IntegerRing(11)) of order 2^4 * 3 * 5^2 * 11 Generators: [2 0] [0 1] [1 1] [0 1] [ 0 1] [10 0] > Group(W) eq GL(2, Integers(11)); true > G, alpha, L, rho := WeilRepresentation(pi);This gives the Weil representation attached to πup to multiplication by an unramified twist. This consists of a permutation group G, an isomorphism alpha identifying G with the automorphisms of the local field L/Qp, and a 2-dimensional G-module rho.
> G; Permutation group G acting on a set of cardinality 6 Order = 6 = 2 * 3 Id(G) (1, 2, 4)(3, 6, 5) (1, 3)(2, 5)(4, 6) > bool := IsIsomorphic(G, DihedralGroup(3)); > bool; true > L; Totally ramified extension defined by the polynomial x^3 + 11 over Unramified extension defined by the polynomial x^2 + 7*x + 2 over 11-adic field mod 11^10
> S := CuspidalSubspace(ModularSymbols(Gamma0(27), 4, 1)); > newforms := NewformDecomposition(S); > Eigenform(newforms[1], 14); q + 3*q^2 + q^4 + 15*q^5 - 25*q^7 - 21*q^8 + 45*q^10 - 15*q^11 + 20*q^13 + O(q^14) > pi:=LocalComponent(newforms[1], 3); > pi; Supercuspidal Representation of GL(2,Q_3) > W:=CuspidalInducingDatum(pi); > W; GModule W of dimension 2 over Rational Field > Group(W); MatrixGroup(2, IntegerRing(9)) of order 2^2 * 3^5 Generators: [1 1] [0 1] [2 0] [0 1] [1 0] [0 2] [1 0] [3 1]These matrices generate (topologically) the Iwahori subgroup of GL2(Z3) consisting of matrices which are upper-triangular modulo 3. W is an irreducible two-dimensional G-module. The representation πis induced from some extension of W to the normalizer of the Iwahori in GL2(Q3).
> E, chi:=AdmissiblePair(pi); > E; Totally ramified extension defined by the polynomial x^2 - 3 over 3-adic ring mod 3^10 > E.1^2; 3 > chi(1+E.1); zeta_3Note that chi can only be evaluated on units of E, so that chi(E.1) would result in an error.
> G, alpha, L, rho := WeilRepresentation(pi); > L; Totally ramified extension defined by the polynomial x^6 - 12*x^4 - 12*x^2 - 3 over Unramified extension defined by the polynomial x + 2 over 3-adic field mod 3^10[Next][Prev] [Right] [Left] [Up] [Index] [Root]