Attributes of Admissible Representations

CentralCharacter(pi) : RepLoc -> GrpDrchElt

The central character of π, where π is an admissible representation on GL(Q_p). This is a Dirichlet character of p-power conductor.

Conductor(pi) : RepLoc -> RngIntElt

The conductor of π, written multiplicatively.

DefiningModularSymbolsSpace(pi) : RepLoc -> ModSym

The space of modular symbols from which π was created.

IsMinimal(pi) : RepLoc -> BoolElt, GrpDrchElt, RepLoc

Given a representation π of GL₂(Q_p), returns true if the conductor of π cannot be lowered by twisting by a character of Q_p ^x . If π is not minimal, the function also returns a minimal representation π^prime together with a Dirichlet character χ, such that π is the twist of π^prime by χ.

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

> 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
> Conductor(pi);
11
> IsTrivial(CentralCharacter(pi));
true