Examples

Examples#

Example: example1#

We consider a newform of weight \(5\) and level \(7\), whose local representation at \(7\) is principal series. These are Dirichlet characters on \({\mathbb{Z}}/7{\mathbb{Z}}\) (the trivial character and the character of order \(2\)): The principal series representation \(\pi\) is the induction up to \({\operatorname{GL}}_2({\mathbb{Q}}_7)\) of a character of the Borel subgroup inflated from a character of the diagonal group \({\mathbb{Q}}_7^{\times} \times {\mathbb{Q}}_7^{\times}.\) The restriction of this character to \({\mathbb{Z}}_7^{\times} \times {\mathbb{Z}}_7^{\times}\) gives the pair of Dirichlet characters above. We now compute the Galois representation. The Weil representation is simply the sum of the two characters above (up to unramified twists), considered as characters of the Galois group of \({\mathbb{Q}}_7\) via local class field theory.

> 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
%%a> assert $1 eq 7;
> parameters := PrincipalSeriesParameters(pi);
> Conductor(parameters[1]), Order(parameters[1]);
1 1
> Conductor(parameters[2]), Order(parameters[2]);
7 2
> rho := WeilRepresentation(pi); rho;
2-dim Galois representation (2,0) with G=C2, I=C2, conductor 7^1 over Q7[40]
> IsAbelian(Group(rho));
true
%%a> assert $1;
> Decomposition(rho);
[
1-dim trivial Galois representation 1 over Q7[40],
1-dim Galois representation (1,-1) with G=C2, I=C2, conductor 7^1 over Q7[40]
]
%%a> assert #$1 eq 2;

Example: example2#

We consider a supercuspidal representation of conductor \(121\), associated to a newform of weight \(2\) and level \(121.\) This means the representation of the Weil group associated to pi is irreducible. \(W\) is a module over a group which is a quotient of \({\operatorname{GL}}_2({\mathbb{Z}}_{11}),\) namely \({\operatorname{GL}}_2({\mathbb{Z}}/{11{\mathbb{Z}}}).\) The representation \(\pi\) is induced from some extension of \(W\) to the open subgroup \({\mathbb{Q}}_{11}^{\times}{\operatorname{GL}}_2({\mathbb{Z}}_{11}).\) This gives the Weil representation attached to \(\pi\) up to multiplication by an unramified twist.

> 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 Rational Field,
Modular symbols space for Gamma_0(121) of weight 2 and dimension 1
   over Rational Field,
Modular symbols space for Gamma_0(121) of weight 2 and dimension 1
   over Rational Field,
Modular symbols space for Gamma_0(121) of weight 2 and dimension 1
   over Rational Field,
Modular symbols space for Gamma_0(121) of weight 2 and dimension 2
   over Rational Field
]
> 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)
> Conductor(pi);
121
%%a> assert $1 eq 121;
> W := CuspidalInducingDatum(pi);
> W;
GModule W of dimension 10 over Rational Field
> 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
%%a> assert $1;
> rho:=WeilRepresentation(pi);
> rho;
2-dim Galois representation (2,0,-1) with G=S3, I=C3, conductor 11^2
   over Q11[10]

Example: example3#

We consider a supercuspidal representation of conductor \(3^3\), associated to a newform of weight \(4\) and level \(27\). These matrices generate (topologically) the Iwahori subgroup of \({\operatorname{GL}}_2({\mathbb{Z}}_3)\) consisting of matrices which are upper-triangular modulo \(3\). \(W\) is an irreducible two-dimensional \(G\)-module. The representation \(\pi\) is induced from some extension of \(W\) to the normalizer of the Iwahori in \({\operatorname{GL}}_2({\mathbb{Q}}_3)\). Note that chi can only be evaluated on units of \(E\), so that chi(E.1) would result in an error.

> S := CuspidalSubspace(ModularSymbols(Gamma0(27), 4, 1));
> newforms := NewformDecomposition(S);
> Eigenform(newforms[1], 13);
q + 3*q^2 + q^4 + 15*q^5 - 25*q^7 - 21*q^8 + 45*q^10 - 15*q^11 + O(q^13)
> 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]
> 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
%%a> assert $1 eq 3;
> chi(1+E.1);
-zeta_3 - 1
> WeilRepresentation(pi);
2-dim Galois representation (2,0,-1) with G=S3, I=S3, conductor 3^3 over Q3[10]

Examples

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Examples#