We　study　genuine　multipartite　entanglement　of　arbitrary　n-partite　quantum　states　by　representing　the　density　matrices　in　terms　of　generalized　Pauli　operators.　While　the　usual　Bloch　representation　of　a　density　matrix　uses　three　types　of　generators　in　the　special　unitary　Lie　algebra　su(d),　the　Weyl　representation　with　generalized　Pauli　operators　has　one　uniformed　type　of　generators　that　simplifies　computation.　In　this　paper,　we　take　the　advantage　of　this　simplicity　to　derive　useful　and　operational　criteria　to　detect　genuine　tripartite　entanglement.　We　also　generalize　the　results　to　obtain　a　sufficient　criterion　to　detect　genuine　entanglement　for　multipartite　quantum　states　in　arbitrary　dimensions.　The　new　method　can　detect　more　genuine　entangled　states　than　previous　methods　as　backed　by　detailed　examples.