Attributes of Admissible Representations

CentralCharacter(pi): RepLoc → GrpDrchElt

The central character of \(\pi\), where \(\pi\) is an admissible representation on \({\operatorname{GL}}({\mathbb{Q}}_p).\) This is a Dirichlet character of \(p\)-power conductor.

Conductor(pi): RepLoc → RngIntElt

The conductor of \(\pi\), written multiplicatively.

DefiningModularSymbolsSpace(pi): RepLoc → ModSym

The space of modular symbols from which \(\pi\) was created.

IsMinimal(pi): RepLoc → BoolElt, GrpDrchElt, RepLoc

Given a representation \(\pi\) of \({\operatorname{GL}}_2({\mathbb{Q}}_p)\), returns true if the conductor of \(\pi\) cannot be lowered by twisting by a character of \({\mathbb{Q}}_p^{\times}.\) If \(\pi\) is not minimal, the function also returns a minimal representation \(\pi^{\prime}\) together with a Dirichlet character \(\chi,\) such that \(\pi\) is the twist of \(\pi^{\prime}\) by \(\chi\).

This is true iff IsMinimalTwist(DefiningModularSymbolsSpace(pi)) is true.

Example: Attributes Example

We continue the previous example.

> S11 := CuspidalSubspace(ModularSymbols(11, 2, 1));
> E11 := NewformDecomposition(S11)[1];
> E11;
Modular symbols space for Gamma_0(11) of weight 2 and dimension 1
   over Rational Field
> pi := LocalComponent(E11, 11);
> pi;
Steinberg Representation of GL(2,Q_11)
> DefiningModularSymbolsSpace(pi) eq E11;
true
%%a> assert $1;
> Conductor(pi);
11
%%a> assert $1 eq 11;
> IsTrivial(CentralCharacter(pi));
true
%%a> assert $1;