E Representations in Point Group C19h

ε=exp(2πi/19)
C_19h		E	C₁₉	(C₁₉)²	(C₁₉)³	(C₁₉)⁴	(C₁₉)⁵	(C₁₉)⁶	(C₁₉)⁷	(C₁₉)⁸	(C₁₉)⁹	(C₁₉)¹⁰	(C₁₉)¹¹	(C₁₉)¹²	(C₁₉)¹³	(C₁₉)¹⁴	(C₁₉)¹⁵	(C₁₉)¹⁶	(C₁₉)¹⁷	(C₁₉)¹⁸	σ_h	S₁₉	(S₁₉)²¹	(S₁₉)³	(S₁₉)²³	(S₁₉)⁵	(S₁₉)²⁵	(S₁₉)⁷	(S₁₉)²⁷	(S₁₉)⁹	(S₁₉)²⁹	(S₁₉)¹¹	(S₁₉)³¹	(S₁₉)¹³	(S₁₉)³³	(S₁₉)¹⁵	(S₁₉)³⁵	(S₁₉)¹⁷	(S₁₉)³⁷
	A'	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1
E'₁	E'_1a E'_1b	1 1	ε^* ε^*	ε^2* ε^2*	ε^3* ε^3*	ε^4* ε^4*	ε^5* ε^5*	ε^6* ε^6*	ε^7* ε^7*	ε^8* ε^8*	ε^9* ε^9*	ε^9* ε^9*	ε^8* ε^8*	ε^7* ε^7*	ε^6* ε^6*	ε^5* ε^5*	ε^4* ε^4*	ε^3* ε^3*	ε^2* ε^2*	ε^* ε^*	1 1	ε^* ε^*	ε^2* ε^2*	ε^3* ε^3*	ε^4* ε^4*	ε^5* ε^5*	ε^6* ε^6*	ε^7* ε^7*	ε^8* ε^8*	ε^9* ε^9*	ε^9* ε^9*	ε^8* ε^8*	ε^7* ε^7*	ε^6* ε^6*	ε^5* ε^5*	ε^4* ε^4*	ε^3* ε^3*	ε^2* ε^2*	ε^* ε^*
E'₂	E'_2a E'_2b	1 1	ε^2* ε^2*	ε^4* ε^4*	ε^6* ε^6*	ε^8* ε^8*	ε^9* ε^9*	ε^7* ε^7*	ε^5* ε^5*	ε^3* ε^3*	ε^* ε^*	ε^* ε^*	ε^3* ε^3*	ε^5* ε^5*	ε^7* ε^7*	ε^9* ε^9*	ε^8* ε^8*	ε^6* ε^6*	ε^4* ε^4*	ε^2* ε^2*	1 1	ε^2* ε^2*	ε^4* ε^4*	ε^6* ε^6*	ε^8* ε^8*	ε^9* ε^9*	ε^7* ε^7*	ε^5* ε^5*	ε^3* ε^3*	ε^* ε^*	ε^* ε^*	ε^3* ε^3*	ε^5* ε^5*	ε^7* ε^7*	ε^9* ε^9*	ε^8* ε^8*	ε^6* ε^6*	ε^4* ε^4*	ε^2* ε^2*
E'₃	E'_3a E'_3b	1 1	ε^3* ε^3*	ε^6* ε^6*	ε^9* ε^9*	ε^7* ε^7*	ε^4* ε^4*	ε^* ε^*	ε^2* ε^2*	ε^5* ε^5*	ε^8* ε^8*	ε^8* ε^8*	ε^5* ε^5*	ε^2* ε^2*	ε^* ε^*	ε^4* ε^4*	ε^7* ε^7*	ε^9* ε^9*	ε^6* ε^6*	ε^3* ε^3*	1 1	ε^3* ε^3*	ε^6* ε^6*	ε^9* ε^9*	ε^7* ε^7*	ε^4* ε^4*	ε^* ε^*	ε^2* ε^2*	ε^5* ε^5*	ε^8* ε^8*	ε^8* ε^8*	ε^5* ε^5*	ε^2* ε^2*	ε^* ε^*	ε^4* ε^4*	ε^7* ε^7*	ε^9* ε^9*	ε^6* ε^6*	ε^3* ε^3*
E'₄	E'_4a E'_4b	1 1	ε^4* ε^4*	ε^8* ε^8*	ε^7* ε^7*	ε^3* ε^3*	ε^* ε^*	ε^5* ε^5*	ε^9* ε^9*	ε^6* ε^6*	ε^2* ε^2*	ε^2* ε^2*	ε^6* ε^6*	ε^9* ε^9*	ε^5* ε^5*	ε^* ε^*	ε^3* ε^3*	ε^7* ε^7*	ε^8* ε^8*	ε^4* ε^4*	1 1	ε^4* ε^4*	ε^8* ε^8*	ε^7* ε^7*	ε^3* ε^3*	ε^* ε^*	ε^5* ε^5*	ε^9* ε^9*	ε^6* ε^6*	ε^2* ε^2*	ε^2* ε^2*	ε^6* ε^6*	ε^9* ε^9*	ε^5* ε^5*	ε^* ε^*	ε^3* ε^3*	ε^7* ε^7*	ε^8* ε^8*	ε^4* ε^4*
E'₅	E'_5a E'_5b	1 1	ε^5* ε^5*	ε^9* ε^9*	ε^4* ε^4*	ε^* ε^*	ε^6* ε^6*	ε^8* ε^8*	ε^3* ε^3*	ε^2* ε^2*	ε^7* ε^7*	ε^7* ε^7*	ε^2* ε^2*	ε^3* ε^3*	ε^8* ε^8*	ε^6* ε^6*	ε^* ε^*	ε^4* ε^4*	ε^9* ε^9*	ε^5* ε^5*	1 1	ε^5* ε^5*	ε^9* ε^9*	ε^4* ε^4*	ε^* ε^*	ε^6* ε^6*	ε^8* ε^8*	ε^3* ε^3*	ε^2* ε^2*	ε^7* ε^7*	ε^7* ε^7*	ε^2* ε^2*	ε^3* ε^3*	ε^8* ε^8*	ε^6* ε^6*	ε^* ε^*	ε^4* ε^4*	ε^9* ε^9*	ε^5* ε^5*
E'₆	E'_6a E'_6b	1 1	ε^6* ε^6*	ε^7* ε^7*	ε^* ε^*	ε^5* ε^5*	ε^8* ε^8*	ε^2* ε^2*	ε^4* ε^4*	ε^9* ε^9*	ε^3* ε^3*	ε^3* ε^3*	ε^9* ε^9*	ε^4* ε^4*	ε^2* ε^2*	ε^8* ε^8*	ε^5* ε^5*	ε^* ε^*	ε^7* ε^7*	ε^6* ε^6*	1 1	ε^6* ε^6*	ε^7* ε^7*	ε^* ε^*	ε^5* ε^5*	ε^8* ε^8*	ε^2* ε^2*	ε^4* ε^4*	ε^9* ε^9*	ε^3* ε^3*	ε^3* ε^3*	ε^9* ε^9*	ε^4* ε^4*	ε^2* ε^2*	ε^8* ε^8*	ε^5* ε^5*	ε^* ε^*	ε^7* ε^7*	ε^6* ε^6*
E'₇	E'_7a E'_7b	1 1	ε^7* ε^7*	ε^5* ε^5*	ε^2* ε^2*	ε^9* ε^9*	ε^3* ε^3*	ε^4* ε^4*	ε^8* ε^8*	ε^* ε^*	ε^6* ε^6*	ε^6* ε^6*	ε^* ε^*	ε^8* ε^8*	ε^4* ε^4*	ε^3* ε^3*	ε^9* ε^9*	ε^2* ε^2*	ε^5* ε^5*	ε^7* ε^7*	1 1	ε^7* ε^7*	ε^5* ε^5*	ε^2* ε^2*	ε^9* ε^9*	ε^3* ε^3*	ε^4* ε^4*	ε^8* ε^8*	ε^* ε^*	ε^6* ε^6*	ε^6* ε^6*	ε^* ε^*	ε^8* ε^8*	ε^4* ε^4*	ε^3* ε^3*	ε^9* ε^9*	ε^2* ε^2*	ε^5* ε^5*	ε^7* ε^7*
E'₈	E'_8a E'_8b	1 1	ε^8* ε^8*	ε^3* ε^3*	ε^5* ε^5*	ε^6* ε^6*	ε^2* ε^2*	ε^9* ε^9*	ε^* ε^*	ε^7* ε^7*	ε^4* ε^4*	ε^4* ε^4*	ε^7* ε^7*	ε^* ε^*	ε^9* ε^9*	ε^2* ε^2*	ε^6* ε^6*	ε^5* ε^5*	ε^3* ε^3*	ε^8* ε^8*	1 1	ε^8* ε^8*	ε^3* ε^3*	ε^5* ε^5*	ε^6* ε^6*	ε^2* ε^2*	ε^9* ε^9*	ε^* ε^*	ε^7* ε^7*	ε^4* ε^4*	ε^4* ε^4*	ε^7* ε^7*	ε^* ε^*	ε^9* ε^9*	ε^2* ε^2*	ε^6* ε^6*	ε^5* ε^5*	ε^3* ε^3*	ε^8* ε^8*
E'₉	E'_9a E'_9b	1 1	ε^9* ε^9*	ε^* ε^*	ε^8* ε^8*	ε^2* ε^2*	ε^7* ε^7*	ε^3* ε^3*	ε^6* ε^6*	ε^4* ε^4*	ε^5* ε^5*	ε^5* ε^5*	ε^4* ε^4*	ε^6* ε^6*	ε^3* ε^3*	ε^7* ε^7*	ε^2* ε^2*	ε^8* ε^8*	ε^* ε^*	ε^9* ε^9*	1 1	ε^9* ε^9*	ε^* ε^*	ε^8* ε^8*	ε^2* ε^2*	ε^7* ε^7*	ε^3* ε^3*	ε^6* ε^6*	ε^4* ε^4*	ε^5* ε^5*	ε^5* ε^5*	ε^4* ε^4*	ε^6* ε^6*	ε^3* ε^3*	ε^7* ε^7*	ε^2* ε^2*	ε^8* ε^8*	ε^* ε^*	ε^9* ε^9*
	A''	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1	-1
E''₁	E''_1a E''_1b	1 1	ε^* ε^*	ε^2* ε^2*	ε^3* ε^3*	ε^4* ε^4*	ε^5* ε^5*	ε^6* ε^6*	ε^7* ε^7*	ε^8* ε^8*	ε^9* ε^9*	ε^9* ε^9*	ε^8* ε^8*	ε^7* ε^7*	ε^6* ε^6*	ε^5* ε^5*	ε^4* ε^4*	ε^3* ε^3*	ε^2* ε^2*	ε^* ε^*	-1 -1	-ε^* -ε^*	-ε^2* -ε^2*	-ε^3* -ε^3*	-ε^4* -ε^4*	-ε^5* -ε^5*	-ε^6* -ε^6*	-ε^7* -ε^7*	-ε^8* -ε^8*	-ε^9* -ε^9*	-ε^9* -ε^9*	-ε^8* -ε^8*	-ε^7* -ε^7*	-ε^6* -ε^6*	-ε^5* -ε^5*	-ε^4* -ε^4*	-ε^3* -ε^3*	-ε^2* -ε^2*	-ε^* -ε^*
E''₂	E''_2a E''_2b	1 1	ε^2* ε^2*	ε^4* ε^4*	ε^6* ε^6*	ε^8* ε^8*	ε^9* ε^9*	ε^7* ε^7*	ε^5* ε^5*	ε^3* ε^3*	ε^* ε^*	ε^* ε^*	ε^3* ε^3*	ε^5* ε^5*	ε^7* ε^7*	ε^9* ε^9*	ε^8* ε^8*	ε^6* ε^6*	ε^4* ε^4*	ε^2* ε^2*	-1 -1	-ε^2* -ε^2*	-ε^4* -ε^4*	-ε^6* -ε^6*	-ε^8* -ε^8*	-ε^9* -ε^9*	-ε^7* -ε^7*	-ε^5* -ε^5*	-ε^3* -ε^3*	-ε^* -ε^*	-ε^* -ε^*	-ε^3* -ε^3*	-ε^5* -ε^5*	-ε^7* -ε^7*	-ε^9* -ε^9*	-ε^8* -ε^8*	-ε^6* -ε^6*	-ε^4* -ε^4*	-ε^2* -ε^2*
E''₃	E''_3a E''_3b	1 1	ε^3* ε^3*	ε^6* ε^6*	ε^9* ε^9*	ε^7* ε^7*	ε^4* ε^4*	ε^* ε^*	ε^2* ε^2*	ε^5* ε^5*	ε^8* ε^8*	ε^8* ε^8*	ε^5* ε^5*	ε^2* ε^2*	ε^* ε^*	ε^4* ε^4*	ε^7* ε^7*	ε^9* ε^9*	ε^6* ε^6*	ε^3* ε^3*	-1 -1	-ε^3* -ε^3*	-ε^6* -ε^6*	-ε^9* -ε^9*	-ε^7* -ε^7*	-ε^4* -ε^4*	-ε^* -ε^*	-ε^2* -ε^2*	-ε^5* -ε^5*	-ε^8* -ε^8*	-ε^8* -ε^8*	-ε^5* -ε^5*	-ε^2* -ε^2*	-ε^* -ε^*	-ε^4* -ε^4*	-ε^7* -ε^7*	-ε^9* -ε^9*	-ε^6* -ε^6*	-ε^3* -ε^3*
E''₄	E''_4a E''_4b	1 1	ε^4* ε^4*	ε^8* ε^8*	ε^7* ε^7*	ε^3* ε^3*	ε^* ε^*	ε^5* ε^5*	ε^9* ε^9*	ε^6* ε^6*	ε^2* ε^2*	ε^2* ε^2*	ε^6* ε^6*	ε^9* ε^9*	ε^5* ε^5*	ε^* ε^*	ε^3* ε^3*	ε^7* ε^7*	ε^8* ε^8*	ε^4* ε^4*	-1 -1	-ε^4* -ε^4*	-ε^8* -ε^8*	-ε^7* -ε^7*	-ε^3* -ε^3*	-ε^* -ε^*	-ε^5* -ε^5*	-ε^9* -ε^9*	-ε^6* -ε^6*	-ε^2* -ε^2*	-ε^2* -ε^2*	-ε^6* -ε^6*	-ε^9* -ε^9*	-ε^5* -ε^5*	-ε^* -ε^*	-ε^3* -ε^3*	-ε^7* -ε^7*	-ε^8* -ε^8*	-ε^4* -ε^4*
E''₅	E''_5a E''_5b	1 1	ε^5* ε^5*	ε^9* ε^9*	ε^4* ε^4*	ε^* ε^*	ε^6* ε^6*	ε^8* ε^8*	ε^3* ε^3*	ε^2* ε^2*	ε^7* ε^7*	ε^7* ε^7*	ε^2* ε^2*	ε^3* ε^3*	ε^8* ε^8*	ε^6* ε^6*	ε^* ε^*	ε^4* ε^4*	ε^9* ε^9*	ε^5* ε^5*	-1 -1	-ε^5* -ε^5*	-ε^9* -ε^9*	-ε^4* -ε^4*	-ε^* -ε^*	-ε^6* -ε^6*	-ε^8* -ε^8*	-ε^3* -ε^3*	-ε^2* -ε^2*	-ε^7* -ε^7*	-ε^7* -ε^7*	-ε^2* -ε^2*	-ε^3* -ε^3*	-ε^8* -ε^8*	-ε^6* -ε^6*	-ε^* -ε^*	-ε^4* -ε^4*	-ε^9* -ε^9*	-ε^5* -ε^5*
E''₆	E''_6a E''_6b	1 1	ε^6* ε^6*	ε^7* ε^7*	ε^* ε^*	ε^5* ε^5*	ε^8* ε^8*	ε^2* ε^2*	ε^4* ε^4*	ε^9* ε^9*	ε^3* ε^3*	ε^3* ε^3*	ε^9* ε^9*	ε^4* ε^4*	ε^2* ε^2*	ε^8* ε^8*	ε^5* ε^5*	ε^* ε^*	ε^7* ε^7*	ε^6* ε^6*	-1 -1	-ε^6* -ε^6*	-ε^7* -ε^7*	-ε^* -ε^*	-ε^5* -ε^5*	-ε^8* -ε^8*	-ε^2* -ε^2*	-ε^4* -ε^4*	-ε^9* -ε^9*	-ε^3* -ε^3*	-ε^3* -ε^3*	-ε^9* -ε^9*	-ε^4* -ε^4*	-ε^2* -ε^2*	-ε^8* -ε^8*	-ε^5* -ε^5*	-ε^* -ε^*	-ε^7* -ε^7*	-ε^6* -ε^6*
E''₇	E''_7a E''_7b	1 1	ε^7* ε^7*	ε^5* ε^5*	ε^2* ε^2*	ε^9* ε^9*	ε^3* ε^3*	ε^4* ε^4*	ε^8* ε^8*	ε^* ε^*	ε^6* ε^6*	ε^6* ε^6*	ε^* ε^*	ε^8* ε^8*	ε^4* ε^4*	ε^3* ε^3*	ε^9* ε^9*	ε^2* ε^2*	ε^5* ε^5*	ε^7* ε^7*	-1 -1	-ε^7* -ε^7*	-ε^5* -ε^5*	-ε^2* -ε^2*	-ε^9* -ε^9*	-ε^3* -ε^3*	-ε^4* -ε^4*	-ε^8* -ε^8*	-ε^* -ε^*	-ε^6* -ε^6*	-ε^6* -ε^6*	-ε^* -ε^*	-ε^8* -ε^8*	-ε^4* -ε^4*	-ε^3* -ε^3*	-ε^9* -ε^9*	-ε^2* -ε^2*	-ε^5* -ε^5*	-ε^7* -ε^7*
E''₈	E''_8a E''_8b	1 1	ε^8* ε^8*	ε^3* ε^3*	ε^5* ε^5*	ε^6* ε^6*	ε^2* ε^2*	ε^9* ε^9*	ε^* ε^*	ε^7* ε^7*	ε^4* ε^4*	ε^4* ε^4*	ε^7* ε^7*	ε^* ε^*	ε^9* ε^9*	ε^2* ε^2*	ε^6* ε^6*	ε^5* ε^5*	ε^3* ε^3*	ε^8* ε^8*	-1 -1	-ε^8* -ε^8*	-ε^3* -ε^3*	-ε^5* -ε^5*	-ε^6* -ε^6*	-ε^2* -ε^2*	-ε^9* -ε^9*	-ε^* -ε^*	-ε^7* -ε^7*	-ε^4* -ε^4*	-ε^4* -ε^4*	-ε^7* -ε^7*	-ε^* -ε^*	-ε^9* -ε^9*	-ε^2* -ε^2*	-ε^6* -ε^6*	-ε^5* -ε^5*	-ε^3* -ε^3*	-ε^8* -ε^8*
E''₉	E''_9a E''_9b	1 1	ε^9* ε^9*	ε^* ε^*	ε^8* ε^8*	ε^2* ε^2*	ε^7* ε^7*	ε^3* ε^3*	ε^6* ε^6*	ε^4* ε^4*	ε^5* ε^5*	ε^5* ε^5*	ε^4* ε^4*	ε^6* ε^6*	ε^3* ε^3*	ε^7* ε^7*	ε^2* ε^2*	ε^8* ε^8*	ε^* ε^*	ε^9* ε^9*	-1 -1	-ε^9* -ε^9*	-ε^* -ε^*	-ε^8* -ε^8*	-ε^2* -ε^2*	-ε^7* -ε^7*	-ε^3* -ε^3*	-ε^6* -ε^6*	-ε^4* -ε^4*	-ε^5* -ε^5*	-ε^5* -ε^5*	-ε^4* -ε^4*	-ε^6* -ε^6*	-ε^3* -ε^3*	-ε^7* -ε^7*	-ε^2* -ε^2*	-ε^8* -ε^8*	-ε^* -ε^*	-ε^9* -ε^9*

Note on E representations in
C_19h character table

h = ∑ Χ_i(E) Χ_i(E)

c_i = 1/h ∑_R n_R χ_i(R)^irred χ(R)^red

c_i(E) = 1/(2h) ∑_R n_R χ_i(R)^irred χ(R)^red = (∑_R n_R χ_i(R)^irred χ(R)^red) / (∑_R n_R χ_i(R)^irred χ_i(R)^irred)

Note on E representations inC19h character table

h = ∑ Χi(E) Χi(E)

ci = 1/h ∑R nR χi(R)irred χ(R)red

ci(E) = 1/(2h) ∑R nR χi(R)irred χ(R)red = (∑R nR χi(R)irred χ(R)red) / (∑R nR χi(R)irred χi(R)irred)

Note on E representations in
C_19h character table

h = ∑ Χ_i(E) Χ_i(E)

c_i = 1/h ∑_R n_R χ_i(R)^irred χ(R)^red

c_i(E) = 1/(2h) ∑_R n_R χ_i(R)^irred χ(R)^red = (∑_R n_R χ_i(R)^irred χ(R)^red) / (∑_R n_R χ_i(R)^irred χ_i(R)^irred)