Examples

Example: example1 (ex-e326da)

We consider a newform of weight \(5\) and level \(7\), whose local representation at \(7\) is principal series.

> 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);

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These are Dirichlet characters on \({\mathbb{Z}}/7{\mathbb{Z}}\) (the trivial character and the character of order \(2\)):

> Conductor(parameters[1]), Order(parameters[1]);
1 1
> Conductor(parameters[2]), Order(parameters[2]);
7 2

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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.

> 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

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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.

> 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]
]

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Example: example2 (ex-298b2e)

We consider a supercuspidal representation of conductor \(121\), associated to a newform of weight \(2\) and level \(121.\)

> 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)

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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 Field

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\(W\) 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 \({\mathbb{Q}}_{11}^{\times}{\operatorname{GL}}_2({\mathbb{Z}}_{11}).\)

> 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
> rho:=WeilRepresentation(pi);

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This gives the full Weil representation attached to \(\pi\). The leading factor Unr(-I) is the unramified twist by \(-1\) from the Loeffler-Weinstein rectifier, which fixes the action of Frobenius.

> rho;
2-dim Galois representation Unr(-I)*(2,0,-1) with G=S3, I=C3, conductor 11^2
   over Q11[10]

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Since the Frobenius action is now determined, the local root number is available.

> RootNumber(rho);
-1

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Example: example3 (ex-3e3aa7)

We consider a supercuspidal representation of conductor \(3^3\), associated to a newform of weight \(4\) and level \(27\).

> 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]

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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 \({\operatorname{GL}}_2({\mathbb{Q}}_3)\).

> 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_3 - 1

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Note that chi can only be evaluated on units of \(E\), so that chi(E.1) would result in an error.

> WeilRepresentation(pi);
2-dim Galois representation Unr(-1/3*sqrt(3)*i)*(2,0,-1) with G=S3, I=S3, conductor 3^3 over Q3[10]

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