Admissible Representations of \({\operatorname{GL}}_2({\mathbb{Q}}_p)\)# Introduction Motivation Definitions The Principal Series Supercuspidal Representations The Local Langlands Correspondence Connection with Modular Forms Category Verbose Output Creation of Admissible Representations LocalComponent(M, p): ModSym, RngIntElt → RepLoc Example: Local Component at 11 of a Newform Attributes of Admissible Representations CentralCharacter(pi): RepLoc → GrpDrchElt Conductor(pi): RepLoc → RngIntElt DefiningModularSymbolsSpace(pi): RepLoc → ModSym IsMinimal(pi): RepLoc → BoolElt, GrpDrchElt, RepLoc Example: Attributes of a Local Representation Structure of Admissible Representations IsPrincipalSeries(pi): RepLoc → BoolElt IsSupercuspidal(pi): RepLoc → BoolElt PrincipalSeriesParameters(pi): RepLoc → GrpDrchElt, GrpDrchElt CuspidalInducingDatum(pi): RepLoc → ModGrp Local Galois Representations GaloisRepresentation(pi): RepLoc → GalRep WeilRepresentation(pi): RepLoc → GalRep AdmissiblePair(pi): RepLoc → RngPad, Map Examples Example: Principal Series Representation at 7 Example: Supercuspidal Representation of Conductor 121 Example: Supercuspidal Representation of Conductor 27 Bibliography