# Properties

 Modulus 179 Structure $$C_{178}$$ Order 178

Show commands for: SageMath / Pari/GP

sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed

sage: H = DirichletGroup_conrey(179)

pari: g = idealstar(,179,2)

## Character group

 sage: G.order()  pari: g.no Order = 178 sage: H.invariants()  pari: g.cyc Structure = $$C_{178}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{179}(2,\cdot)$

## First 32 of 178 characters

Each row describes a character. When available, the columns show the orbit label, order of the character, whether the character is primitive, and several values of the character.

orbit label order primitive -1 1 2 3 4 5 6 7 8 9 10 11
$$\chi_{179}(1,\cdot)$$ 179.a 1 No $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{179}(2,\cdot)$$ 179.d 178 Yes $$-1$$ $$1$$ $$e\left(\frac{1}{178}\right)$$ $$e\left(\frac{54}{89}\right)$$ $$e\left(\frac{1}{89}\right)$$ $$e\left(\frac{69}{89}\right)$$ $$e\left(\frac{109}{178}\right)$$ $$e\left(\frac{171}{178}\right)$$ $$e\left(\frac{3}{178}\right)$$ $$e\left(\frac{19}{89}\right)$$ $$e\left(\frac{139}{178}\right)$$ $$e\left(\frac{15}{178}\right)$$
$$\chi_{179}(3,\cdot)$$ 179.c 89 Yes $$1$$ $$1$$ $$e\left(\frac{54}{89}\right)$$ $$e\left(\frac{47}{89}\right)$$ $$e\left(\frac{19}{89}\right)$$ $$e\left(\frac{65}{89}\right)$$ $$e\left(\frac{12}{89}\right)$$ $$e\left(\frac{67}{89}\right)$$ $$e\left(\frac{73}{89}\right)$$ $$e\left(\frac{5}{89}\right)$$ $$e\left(\frac{30}{89}\right)$$ $$e\left(\frac{9}{89}\right)$$
$$\chi_{179}(4,\cdot)$$ 179.c 89 Yes $$1$$ $$1$$ $$e\left(\frac{1}{89}\right)$$ $$e\left(\frac{19}{89}\right)$$ $$e\left(\frac{2}{89}\right)$$ $$e\left(\frac{49}{89}\right)$$ $$e\left(\frac{20}{89}\right)$$ $$e\left(\frac{82}{89}\right)$$ $$e\left(\frac{3}{89}\right)$$ $$e\left(\frac{38}{89}\right)$$ $$e\left(\frac{50}{89}\right)$$ $$e\left(\frac{15}{89}\right)$$
$$\chi_{179}(5,\cdot)$$ 179.c 89 Yes $$1$$ $$1$$ $$e\left(\frac{69}{89}\right)$$ $$e\left(\frac{65}{89}\right)$$ $$e\left(\frac{49}{89}\right)$$ $$e\left(\frac{88}{89}\right)$$ $$e\left(\frac{45}{89}\right)$$ $$e\left(\frac{51}{89}\right)$$ $$e\left(\frac{29}{89}\right)$$ $$e\left(\frac{41}{89}\right)$$ $$e\left(\frac{68}{89}\right)$$ $$e\left(\frac{56}{89}\right)$$
$$\chi_{179}(6,\cdot)$$ 179.d 178 Yes $$-1$$ $$1$$ $$e\left(\frac{109}{178}\right)$$ $$e\left(\frac{12}{89}\right)$$ $$e\left(\frac{20}{89}\right)$$ $$e\left(\frac{45}{89}\right)$$ $$e\left(\frac{133}{178}\right)$$ $$e\left(\frac{127}{178}\right)$$ $$e\left(\frac{149}{178}\right)$$ $$e\left(\frac{24}{89}\right)$$ $$e\left(\frac{21}{178}\right)$$ $$e\left(\frac{33}{178}\right)$$
$$\chi_{179}(7,\cdot)$$ 179.d 178 Yes $$-1$$ $$1$$ $$e\left(\frac{171}{178}\right)$$ $$e\left(\frac{67}{89}\right)$$ $$e\left(\frac{82}{89}\right)$$ $$e\left(\frac{51}{89}\right)$$ $$e\left(\frac{127}{178}\right)$$ $$e\left(\frac{49}{178}\right)$$ $$e\left(\frac{157}{178}\right)$$ $$e\left(\frac{45}{89}\right)$$ $$e\left(\frac{95}{178}\right)$$ $$e\left(\frac{73}{178}\right)$$
$$\chi_{179}(8,\cdot)$$ 179.d 178 Yes $$-1$$ $$1$$ $$e\left(\frac{3}{178}\right)$$ $$e\left(\frac{73}{89}\right)$$ $$e\left(\frac{3}{89}\right)$$ $$e\left(\frac{29}{89}\right)$$ $$e\left(\frac{149}{178}\right)$$ $$e\left(\frac{157}{178}\right)$$ $$e\left(\frac{9}{178}\right)$$ $$e\left(\frac{57}{89}\right)$$ $$e\left(\frac{61}{178}\right)$$ $$e\left(\frac{45}{178}\right)$$
$$\chi_{179}(9,\cdot)$$ 179.c 89 Yes $$1$$ $$1$$ $$e\left(\frac{19}{89}\right)$$ $$e\left(\frac{5}{89}\right)$$ $$e\left(\frac{38}{89}\right)$$ $$e\left(\frac{41}{89}\right)$$ $$e\left(\frac{24}{89}\right)$$ $$e\left(\frac{45}{89}\right)$$ $$e\left(\frac{57}{89}\right)$$ $$e\left(\frac{10}{89}\right)$$ $$e\left(\frac{60}{89}\right)$$ $$e\left(\frac{18}{89}\right)$$
$$\chi_{179}(10,\cdot)$$ 179.d 178 Yes $$-1$$ $$1$$ $$e\left(\frac{139}{178}\right)$$ $$e\left(\frac{30}{89}\right)$$ $$e\left(\frac{50}{89}\right)$$ $$e\left(\frac{68}{89}\right)$$ $$e\left(\frac{21}{178}\right)$$ $$e\left(\frac{95}{178}\right)$$ $$e\left(\frac{61}{178}\right)$$ $$e\left(\frac{60}{89}\right)$$ $$e\left(\frac{97}{178}\right)$$ $$e\left(\frac{127}{178}\right)$$
$$\chi_{179}(11,\cdot)$$ 179.d 178 Yes $$-1$$ $$1$$ $$e\left(\frac{15}{178}\right)$$ $$e\left(\frac{9}{89}\right)$$ $$e\left(\frac{15}{89}\right)$$ $$e\left(\frac{56}{89}\right)$$ $$e\left(\frac{33}{178}\right)$$ $$e\left(\frac{73}{178}\right)$$ $$e\left(\frac{45}{178}\right)$$ $$e\left(\frac{18}{89}\right)$$ $$e\left(\frac{127}{178}\right)$$ $$e\left(\frac{47}{178}\right)$$
$$\chi_{179}(12,\cdot)$$ 179.c 89 Yes $$1$$ $$1$$ $$e\left(\frac{55}{89}\right)$$ $$e\left(\frac{66}{89}\right)$$ $$e\left(\frac{21}{89}\right)$$ $$e\left(\frac{25}{89}\right)$$ $$e\left(\frac{32}{89}\right)$$ $$e\left(\frac{60}{89}\right)$$ $$e\left(\frac{76}{89}\right)$$ $$e\left(\frac{43}{89}\right)$$ $$e\left(\frac{80}{89}\right)$$ $$e\left(\frac{24}{89}\right)$$
$$\chi_{179}(13,\cdot)$$ 179.c 89 Yes $$1$$ $$1$$ $$e\left(\frac{57}{89}\right)$$ $$e\left(\frac{15}{89}\right)$$ $$e\left(\frac{25}{89}\right)$$ $$e\left(\frac{34}{89}\right)$$ $$e\left(\frac{72}{89}\right)$$ $$e\left(\frac{46}{89}\right)$$ $$e\left(\frac{82}{89}\right)$$ $$e\left(\frac{30}{89}\right)$$ $$e\left(\frac{2}{89}\right)$$ $$e\left(\frac{54}{89}\right)$$
$$\chi_{179}(14,\cdot)$$ 179.c 89 Yes $$1$$ $$1$$ $$e\left(\frac{86}{89}\right)$$ $$e\left(\frac{32}{89}\right)$$ $$e\left(\frac{83}{89}\right)$$ $$e\left(\frac{31}{89}\right)$$ $$e\left(\frac{29}{89}\right)$$ $$e\left(\frac{21}{89}\right)$$ $$e\left(\frac{80}{89}\right)$$ $$e\left(\frac{64}{89}\right)$$ $$e\left(\frac{28}{89}\right)$$ $$e\left(\frac{44}{89}\right)$$
$$\chi_{179}(15,\cdot)$$ 179.c 89 Yes $$1$$ $$1$$ $$e\left(\frac{34}{89}\right)$$ $$e\left(\frac{23}{89}\right)$$ $$e\left(\frac{68}{89}\right)$$ $$e\left(\frac{64}{89}\right)$$ $$e\left(\frac{57}{89}\right)$$ $$e\left(\frac{29}{89}\right)$$ $$e\left(\frac{13}{89}\right)$$ $$e\left(\frac{46}{89}\right)$$ $$e\left(\frac{9}{89}\right)$$ $$e\left(\frac{65}{89}\right)$$
$$\chi_{179}(16,\cdot)$$ 179.c 89 Yes $$1$$ $$1$$ $$e\left(\frac{2}{89}\right)$$ $$e\left(\frac{38}{89}\right)$$ $$e\left(\frac{4}{89}\right)$$ $$e\left(\frac{9}{89}\right)$$ $$e\left(\frac{40}{89}\right)$$ $$e\left(\frac{75}{89}\right)$$ $$e\left(\frac{6}{89}\right)$$ $$e\left(\frac{76}{89}\right)$$ $$e\left(\frac{11}{89}\right)$$ $$e\left(\frac{30}{89}\right)$$
$$\chi_{179}(17,\cdot)$$ 179.c 89 Yes $$1$$ $$1$$ $$e\left(\frac{83}{89}\right)$$ $$e\left(\frac{64}{89}\right)$$ $$e\left(\frac{77}{89}\right)$$ $$e\left(\frac{62}{89}\right)$$ $$e\left(\frac{58}{89}\right)$$ $$e\left(\frac{42}{89}\right)$$ $$e\left(\frac{71}{89}\right)$$ $$e\left(\frac{39}{89}\right)$$ $$e\left(\frac{56}{89}\right)$$ $$e\left(\frac{88}{89}\right)$$
$$\chi_{179}(18,\cdot)$$ 179.d 178 Yes $$-1$$ $$1$$ $$e\left(\frac{39}{178}\right)$$ $$e\left(\frac{59}{89}\right)$$ $$e\left(\frac{39}{89}\right)$$ $$e\left(\frac{21}{89}\right)$$ $$e\left(\frac{157}{178}\right)$$ $$e\left(\frac{83}{178}\right)$$ $$e\left(\frac{117}{178}\right)$$ $$e\left(\frac{29}{89}\right)$$ $$e\left(\frac{81}{178}\right)$$ $$e\left(\frac{51}{178}\right)$$
$$\chi_{179}(19,\cdot)$$ 179.c 89 Yes $$1$$ $$1$$ $$e\left(\frac{27}{89}\right)$$ $$e\left(\frac{68}{89}\right)$$ $$e\left(\frac{54}{89}\right)$$ $$e\left(\frac{77}{89}\right)$$ $$e\left(\frac{6}{89}\right)$$ $$e\left(\frac{78}{89}\right)$$ $$e\left(\frac{81}{89}\right)$$ $$e\left(\frac{47}{89}\right)$$ $$e\left(\frac{15}{89}\right)$$ $$e\left(\frac{49}{89}\right)$$
$$\chi_{179}(20,\cdot)$$ 179.c 89 Yes $$1$$ $$1$$ $$e\left(\frac{70}{89}\right)$$ $$e\left(\frac{84}{89}\right)$$ $$e\left(\frac{51}{89}\right)$$ $$e\left(\frac{48}{89}\right)$$ $$e\left(\frac{65}{89}\right)$$ $$e\left(\frac{44}{89}\right)$$ $$e\left(\frac{32}{89}\right)$$ $$e\left(\frac{79}{89}\right)$$ $$e\left(\frac{29}{89}\right)$$ $$e\left(\frac{71}{89}\right)$$
$$\chi_{179}(21,\cdot)$$ 179.d 178 Yes $$-1$$ $$1$$ $$e\left(\frac{101}{178}\right)$$ $$e\left(\frac{25}{89}\right)$$ $$e\left(\frac{12}{89}\right)$$ $$e\left(\frac{27}{89}\right)$$ $$e\left(\frac{151}{178}\right)$$ $$e\left(\frac{5}{178}\right)$$ $$e\left(\frac{125}{178}\right)$$ $$e\left(\frac{50}{89}\right)$$ $$e\left(\frac{155}{178}\right)$$ $$e\left(\frac{91}{178}\right)$$
$$\chi_{179}(22,\cdot)$$ 179.c 89 Yes $$1$$ $$1$$ $$e\left(\frac{8}{89}\right)$$ $$e\left(\frac{63}{89}\right)$$ $$e\left(\frac{16}{89}\right)$$ $$e\left(\frac{36}{89}\right)$$ $$e\left(\frac{71}{89}\right)$$ $$e\left(\frac{33}{89}\right)$$ $$e\left(\frac{24}{89}\right)$$ $$e\left(\frac{37}{89}\right)$$ $$e\left(\frac{44}{89}\right)$$ $$e\left(\frac{31}{89}\right)$$
$$\chi_{179}(23,\cdot)$$ 179.d 178 Yes $$-1$$ $$1$$ $$e\left(\frac{135}{178}\right)$$ $$e\left(\frac{81}{89}\right)$$ $$e\left(\frac{46}{89}\right)$$ $$e\left(\frac{59}{89}\right)$$ $$e\left(\frac{119}{178}\right)$$ $$e\left(\frac{123}{178}\right)$$ $$e\left(\frac{49}{178}\right)$$ $$e\left(\frac{73}{89}\right)$$ $$e\left(\frac{75}{178}\right)$$ $$e\left(\frac{67}{178}\right)$$
$$\chi_{179}(24,\cdot)$$ 179.d 178 Yes $$-1$$ $$1$$ $$e\left(\frac{111}{178}\right)$$ $$e\left(\frac{31}{89}\right)$$ $$e\left(\frac{22}{89}\right)$$ $$e\left(\frac{5}{89}\right)$$ $$e\left(\frac{173}{178}\right)$$ $$e\left(\frac{113}{178}\right)$$ $$e\left(\frac{155}{178}\right)$$ $$e\left(\frac{62}{89}\right)$$ $$e\left(\frac{121}{178}\right)$$ $$e\left(\frac{63}{178}\right)$$
$$\chi_{179}(25,\cdot)$$ 179.c 89 Yes $$1$$ $$1$$ $$e\left(\frac{49}{89}\right)$$ $$e\left(\frac{41}{89}\right)$$ $$e\left(\frac{9}{89}\right)$$ $$e\left(\frac{87}{89}\right)$$ $$e\left(\frac{1}{89}\right)$$ $$e\left(\frac{13}{89}\right)$$ $$e\left(\frac{58}{89}\right)$$ $$e\left(\frac{82}{89}\right)$$ $$e\left(\frac{47}{89}\right)$$ $$e\left(\frac{23}{89}\right)$$
$$\chi_{179}(26,\cdot)$$ 179.d 178 Yes $$-1$$ $$1$$ $$e\left(\frac{115}{178}\right)$$ $$e\left(\frac{69}{89}\right)$$ $$e\left(\frac{26}{89}\right)$$ $$e\left(\frac{14}{89}\right)$$ $$e\left(\frac{75}{178}\right)$$ $$e\left(\frac{85}{178}\right)$$ $$e\left(\frac{167}{178}\right)$$ $$e\left(\frac{49}{89}\right)$$ $$e\left(\frac{143}{178}\right)$$ $$e\left(\frac{123}{178}\right)$$
$$\chi_{179}(27,\cdot)$$ 179.c 89 Yes $$1$$ $$1$$ $$e\left(\frac{73}{89}\right)$$ $$e\left(\frac{52}{89}\right)$$ $$e\left(\frac{57}{89}\right)$$ $$e\left(\frac{17}{89}\right)$$ $$e\left(\frac{36}{89}\right)$$ $$e\left(\frac{23}{89}\right)$$ $$e\left(\frac{41}{89}\right)$$ $$e\left(\frac{15}{89}\right)$$ $$e\left(\frac{1}{89}\right)$$ $$e\left(\frac{27}{89}\right)$$
$$\chi_{179}(28,\cdot)$$ 179.d 178 Yes $$-1$$ $$1$$ $$e\left(\frac{173}{178}\right)$$ $$e\left(\frac{86}{89}\right)$$ $$e\left(\frac{84}{89}\right)$$ $$e\left(\frac{11}{89}\right)$$ $$e\left(\frac{167}{178}\right)$$ $$e\left(\frac{35}{178}\right)$$ $$e\left(\frac{163}{178}\right)$$ $$e\left(\frac{83}{89}\right)$$ $$e\left(\frac{17}{178}\right)$$ $$e\left(\frac{103}{178}\right)$$
$$\chi_{179}(29,\cdot)$$ 179.c 89 Yes $$1$$ $$1$$ $$e\left(\frac{59}{89}\right)$$ $$e\left(\frac{53}{89}\right)$$ $$e\left(\frac{29}{89}\right)$$ $$e\left(\frac{43}{89}\right)$$ $$e\left(\frac{23}{89}\right)$$ $$e\left(\frac{32}{89}\right)$$ $$e\left(\frac{88}{89}\right)$$ $$e\left(\frac{17}{89}\right)$$ $$e\left(\frac{13}{89}\right)$$ $$e\left(\frac{84}{89}\right)$$
$$\chi_{179}(30,\cdot)$$ 179.d 178 Yes $$-1$$ $$1$$ $$e\left(\frac{69}{178}\right)$$ $$e\left(\frac{77}{89}\right)$$ $$e\left(\frac{69}{89}\right)$$ $$e\left(\frac{44}{89}\right)$$ $$e\left(\frac{45}{178}\right)$$ $$e\left(\frac{51}{178}\right)$$ $$e\left(\frac{29}{178}\right)$$ $$e\left(\frac{65}{89}\right)$$ $$e\left(\frac{157}{178}\right)$$ $$e\left(\frac{145}{178}\right)$$
$$\chi_{179}(31,\cdot)$$ 179.c 89 Yes $$1$$ $$1$$ $$e\left(\frac{31}{89}\right)$$ $$e\left(\frac{55}{89}\right)$$ $$e\left(\frac{62}{89}\right)$$ $$e\left(\frac{6}{89}\right)$$ $$e\left(\frac{86}{89}\right)$$ $$e\left(\frac{50}{89}\right)$$ $$e\left(\frac{4}{89}\right)$$ $$e\left(\frac{21}{89}\right)$$ $$e\left(\frac{37}{89}\right)$$ $$e\left(\frac{20}{89}\right)$$
$$\chi_{179}(32,\cdot)$$ 179.d 178 Yes $$-1$$ $$1$$ $$e\left(\frac{5}{178}\right)$$ $$e\left(\frac{3}{89}\right)$$ $$e\left(\frac{5}{89}\right)$$ $$e\left(\frac{78}{89}\right)$$ $$e\left(\frac{11}{178}\right)$$ $$e\left(\frac{143}{178}\right)$$ $$e\left(\frac{15}{178}\right)$$ $$e\left(\frac{6}{89}\right)$$ $$e\left(\frac{161}{178}\right)$$ $$e\left(\frac{75}{178}\right)$$