# 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 (ex-ad405c)`

We continue the previous example.

```magma
> 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

```