# Properties

 Modulus $173$ Structure $$C_{172}$$ Order $172$

Show commands for: Pari/GP / SageMath

sage: H = DirichletGroup(173)

pari: g = idealstar(,173,2)

## Character group

 sage: G.order()  pari: g.no Order = 172 sage: H.invariants()  pari: g.cyc Structure = $$C_{172}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{173}(2,\cdot)$

## First 32 of 172 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.

Character Orbit Order Primitive $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$
$$\chi_{173}(1,\cdot)$$ 173.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{173}(2,\cdot)$$ 173.f 172 yes $$-1$$ $$1$$ $$e\left(\frac{1}{172}\right)$$ $$e\left(\frac{27}{172}\right)$$ $$e\left(\frac{1}{86}\right)$$ $$e\left(\frac{39}{172}\right)$$ $$e\left(\frac{7}{43}\right)$$ $$e\left(\frac{95}{172}\right)$$ $$e\left(\frac{3}{172}\right)$$ $$e\left(\frac{27}{86}\right)$$ $$e\left(\frac{10}{43}\right)$$ $$e\left(\frac{23}{172}\right)$$
$$\chi_{173}(3,\cdot)$$ 173.f 172 yes $$-1$$ $$1$$ $$e\left(\frac{27}{172}\right)$$ $$e\left(\frac{41}{172}\right)$$ $$e\left(\frac{27}{86}\right)$$ $$e\left(\frac{21}{172}\right)$$ $$e\left(\frac{17}{43}\right)$$ $$e\left(\frac{157}{172}\right)$$ $$e\left(\frac{81}{172}\right)$$ $$e\left(\frac{41}{86}\right)$$ $$e\left(\frac{12}{43}\right)$$ $$e\left(\frac{105}{172}\right)$$
$$\chi_{173}(4,\cdot)$$ 173.e 86 yes $$1$$ $$1$$ $$e\left(\frac{1}{86}\right)$$ $$e\left(\frac{27}{86}\right)$$ $$e\left(\frac{1}{43}\right)$$ $$e\left(\frac{39}{86}\right)$$ $$e\left(\frac{14}{43}\right)$$ $$e\left(\frac{9}{86}\right)$$ $$e\left(\frac{3}{86}\right)$$ $$e\left(\frac{27}{43}\right)$$ $$e\left(\frac{20}{43}\right)$$ $$e\left(\frac{23}{86}\right)$$
$$\chi_{173}(5,\cdot)$$ 173.f 172 yes $$-1$$ $$1$$ $$e\left(\frac{39}{172}\right)$$ $$e\left(\frac{21}{172}\right)$$ $$e\left(\frac{39}{86}\right)$$ $$e\left(\frac{145}{172}\right)$$ $$e\left(\frac{15}{43}\right)$$ $$e\left(\frac{93}{172}\right)$$ $$e\left(\frac{117}{172}\right)$$ $$e\left(\frac{21}{86}\right)$$ $$e\left(\frac{3}{43}\right)$$ $$e\left(\frac{37}{172}\right)$$
$$\chi_{173}(6,\cdot)$$ 173.d 43 yes $$1$$ $$1$$ $$e\left(\frac{7}{43}\right)$$ $$e\left(\frac{17}{43}\right)$$ $$e\left(\frac{14}{43}\right)$$ $$e\left(\frac{15}{43}\right)$$ $$e\left(\frac{24}{43}\right)$$ $$e\left(\frac{20}{43}\right)$$ $$e\left(\frac{21}{43}\right)$$ $$e\left(\frac{34}{43}\right)$$ $$e\left(\frac{22}{43}\right)$$ $$e\left(\frac{32}{43}\right)$$
$$\chi_{173}(7,\cdot)$$ 173.f 172 yes $$-1$$ $$1$$ $$e\left(\frac{95}{172}\right)$$ $$e\left(\frac{157}{172}\right)$$ $$e\left(\frac{9}{86}\right)$$ $$e\left(\frac{93}{172}\right)$$ $$e\left(\frac{20}{43}\right)$$ $$e\left(\frac{81}{172}\right)$$ $$e\left(\frac{113}{172}\right)$$ $$e\left(\frac{71}{86}\right)$$ $$e\left(\frac{4}{43}\right)$$ $$e\left(\frac{121}{172}\right)$$
$$\chi_{173}(8,\cdot)$$ 173.f 172 yes $$-1$$ $$1$$ $$e\left(\frac{3}{172}\right)$$ $$e\left(\frac{81}{172}\right)$$ $$e\left(\frac{3}{86}\right)$$ $$e\left(\frac{117}{172}\right)$$ $$e\left(\frac{21}{43}\right)$$ $$e\left(\frac{113}{172}\right)$$ $$e\left(\frac{9}{172}\right)$$ $$e\left(\frac{81}{86}\right)$$ $$e\left(\frac{30}{43}\right)$$ $$e\left(\frac{69}{172}\right)$$
$$\chi_{173}(9,\cdot)$$ 173.e 86 yes $$1$$ $$1$$ $$e\left(\frac{27}{86}\right)$$ $$e\left(\frac{41}{86}\right)$$ $$e\left(\frac{27}{43}\right)$$ $$e\left(\frac{21}{86}\right)$$ $$e\left(\frac{34}{43}\right)$$ $$e\left(\frac{71}{86}\right)$$ $$e\left(\frac{81}{86}\right)$$ $$e\left(\frac{41}{43}\right)$$ $$e\left(\frac{24}{43}\right)$$ $$e\left(\frac{19}{86}\right)$$
$$\chi_{173}(10,\cdot)$$ 173.d 43 yes $$1$$ $$1$$ $$e\left(\frac{10}{43}\right)$$ $$e\left(\frac{12}{43}\right)$$ $$e\left(\frac{20}{43}\right)$$ $$e\left(\frac{3}{43}\right)$$ $$e\left(\frac{22}{43}\right)$$ $$e\left(\frac{4}{43}\right)$$ $$e\left(\frac{30}{43}\right)$$ $$e\left(\frac{24}{43}\right)$$ $$e\left(\frac{13}{43}\right)$$ $$e\left(\frac{15}{43}\right)$$
$$\chi_{173}(11,\cdot)$$ 173.f 172 yes $$-1$$ $$1$$ $$e\left(\frac{23}{172}\right)$$ $$e\left(\frac{105}{172}\right)$$ $$e\left(\frac{23}{86}\right)$$ $$e\left(\frac{37}{172}\right)$$ $$e\left(\frac{32}{43}\right)$$ $$e\left(\frac{121}{172}\right)$$ $$e\left(\frac{69}{172}\right)$$ $$e\left(\frac{19}{86}\right)$$ $$e\left(\frac{15}{43}\right)$$ $$e\left(\frac{13}{172}\right)$$
$$\chi_{173}(12,\cdot)$$ 173.f 172 yes $$-1$$ $$1$$ $$e\left(\frac{29}{172}\right)$$ $$e\left(\frac{95}{172}\right)$$ $$e\left(\frac{29}{86}\right)$$ $$e\left(\frac{99}{172}\right)$$ $$e\left(\frac{31}{43}\right)$$ $$e\left(\frac{3}{172}\right)$$ $$e\left(\frac{87}{172}\right)$$ $$e\left(\frac{9}{86}\right)$$ $$e\left(\frac{32}{43}\right)$$ $$e\left(\frac{151}{172}\right)$$
$$\chi_{173}(13,\cdot)$$ 173.e 86 yes $$1$$ $$1$$ $$e\left(\frac{65}{86}\right)$$ $$e\left(\frac{35}{86}\right)$$ $$e\left(\frac{22}{43}\right)$$ $$e\left(\frac{41}{86}\right)$$ $$e\left(\frac{7}{43}\right)$$ $$e\left(\frac{69}{86}\right)$$ $$e\left(\frac{23}{86}\right)$$ $$e\left(\frac{35}{43}\right)$$ $$e\left(\frac{10}{43}\right)$$ $$e\left(\frac{33}{86}\right)$$
$$\chi_{173}(14,\cdot)$$ 173.d 43 yes $$1$$ $$1$$ $$e\left(\frac{24}{43}\right)$$ $$e\left(\frac{3}{43}\right)$$ $$e\left(\frac{5}{43}\right)$$ $$e\left(\frac{33}{43}\right)$$ $$e\left(\frac{27}{43}\right)$$ $$e\left(\frac{1}{43}\right)$$ $$e\left(\frac{29}{43}\right)$$ $$e\left(\frac{6}{43}\right)$$ $$e\left(\frac{14}{43}\right)$$ $$e\left(\frac{36}{43}\right)$$
$$\chi_{173}(15,\cdot)$$ 173.e 86 yes $$1$$ $$1$$ $$e\left(\frac{33}{86}\right)$$ $$e\left(\frac{31}{86}\right)$$ $$e\left(\frac{33}{43}\right)$$ $$e\left(\frac{83}{86}\right)$$ $$e\left(\frac{32}{43}\right)$$ $$e\left(\frac{39}{86}\right)$$ $$e\left(\frac{13}{86}\right)$$ $$e\left(\frac{31}{43}\right)$$ $$e\left(\frac{15}{43}\right)$$ $$e\left(\frac{71}{86}\right)$$
$$\chi_{173}(16,\cdot)$$ 173.d 43 yes $$1$$ $$1$$ $$e\left(\frac{1}{43}\right)$$ $$e\left(\frac{27}{43}\right)$$ $$e\left(\frac{2}{43}\right)$$ $$e\left(\frac{39}{43}\right)$$ $$e\left(\frac{28}{43}\right)$$ $$e\left(\frac{9}{43}\right)$$ $$e\left(\frac{3}{43}\right)$$ $$e\left(\frac{11}{43}\right)$$ $$e\left(\frac{40}{43}\right)$$ $$e\left(\frac{23}{43}\right)$$
$$\chi_{173}(17,\cdot)$$ 173.f 172 yes $$-1$$ $$1$$ $$e\left(\frac{73}{172}\right)$$ $$e\left(\frac{79}{172}\right)$$ $$e\left(\frac{73}{86}\right)$$ $$e\left(\frac{95}{172}\right)$$ $$e\left(\frac{38}{43}\right)$$ $$e\left(\frac{55}{172}\right)$$ $$e\left(\frac{47}{172}\right)$$ $$e\left(\frac{79}{86}\right)$$ $$e\left(\frac{42}{43}\right)$$ $$e\left(\frac{131}{172}\right)$$
$$\chi_{173}(18,\cdot)$$ 173.f 172 yes $$-1$$ $$1$$ $$e\left(\frac{55}{172}\right)$$ $$e\left(\frac{109}{172}\right)$$ $$e\left(\frac{55}{86}\right)$$ $$e\left(\frac{81}{172}\right)$$ $$e\left(\frac{41}{43}\right)$$ $$e\left(\frac{65}{172}\right)$$ $$e\left(\frac{165}{172}\right)$$ $$e\left(\frac{23}{86}\right)$$ $$e\left(\frac{34}{43}\right)$$ $$e\left(\frac{61}{172}\right)$$
$$\chi_{173}(19,\cdot)$$ 173.f 172 yes $$-1$$ $$1$$ $$e\left(\frac{33}{172}\right)$$ $$e\left(\frac{31}{172}\right)$$ $$e\left(\frac{33}{86}\right)$$ $$e\left(\frac{83}{172}\right)$$ $$e\left(\frac{16}{43}\right)$$ $$e\left(\frac{39}{172}\right)$$ $$e\left(\frac{99}{172}\right)$$ $$e\left(\frac{31}{86}\right)$$ $$e\left(\frac{29}{43}\right)$$ $$e\left(\frac{71}{172}\right)$$
$$\chi_{173}(20,\cdot)$$ 173.f 172 yes $$-1$$ $$1$$ $$e\left(\frac{41}{172}\right)$$ $$e\left(\frac{75}{172}\right)$$ $$e\left(\frac{41}{86}\right)$$ $$e\left(\frac{51}{172}\right)$$ $$e\left(\frac{29}{43}\right)$$ $$e\left(\frac{111}{172}\right)$$ $$e\left(\frac{123}{172}\right)$$ $$e\left(\frac{75}{86}\right)$$ $$e\left(\frac{23}{43}\right)$$ $$e\left(\frac{83}{172}\right)$$
$$\chi_{173}(21,\cdot)$$ 173.e 86 yes $$1$$ $$1$$ $$e\left(\frac{61}{86}\right)$$ $$e\left(\frac{13}{86}\right)$$ $$e\left(\frac{18}{43}\right)$$ $$e\left(\frac{57}{86}\right)$$ $$e\left(\frac{37}{43}\right)$$ $$e\left(\frac{33}{86}\right)$$ $$e\left(\frac{11}{86}\right)$$ $$e\left(\frac{13}{43}\right)$$ $$e\left(\frac{16}{43}\right)$$ $$e\left(\frac{27}{86}\right)$$
$$\chi_{173}(22,\cdot)$$ 173.d 43 yes $$1$$ $$1$$ $$e\left(\frac{6}{43}\right)$$ $$e\left(\frac{33}{43}\right)$$ $$e\left(\frac{12}{43}\right)$$ $$e\left(\frac{19}{43}\right)$$ $$e\left(\frac{39}{43}\right)$$ $$e\left(\frac{11}{43}\right)$$ $$e\left(\frac{18}{43}\right)$$ $$e\left(\frac{23}{43}\right)$$ $$e\left(\frac{25}{43}\right)$$ $$e\left(\frac{9}{43}\right)$$
$$\chi_{173}(23,\cdot)$$ 173.d 43 yes $$1$$ $$1$$ $$e\left(\frac{5}{43}\right)$$ $$e\left(\frac{6}{43}\right)$$ $$e\left(\frac{10}{43}\right)$$ $$e\left(\frac{23}{43}\right)$$ $$e\left(\frac{11}{43}\right)$$ $$e\left(\frac{2}{43}\right)$$ $$e\left(\frac{15}{43}\right)$$ $$e\left(\frac{12}{43}\right)$$ $$e\left(\frac{28}{43}\right)$$ $$e\left(\frac{29}{43}\right)$$
$$\chi_{173}(24,\cdot)$$ 173.e 86 yes $$1$$ $$1$$ $$e\left(\frac{15}{86}\right)$$ $$e\left(\frac{61}{86}\right)$$ $$e\left(\frac{15}{43}\right)$$ $$e\left(\frac{69}{86}\right)$$ $$e\left(\frac{38}{43}\right)$$ $$e\left(\frac{49}{86}\right)$$ $$e\left(\frac{45}{86}\right)$$ $$e\left(\frac{18}{43}\right)$$ $$e\left(\frac{42}{43}\right)$$ $$e\left(\frac{1}{86}\right)$$
$$\chi_{173}(25,\cdot)$$ 173.e 86 yes $$1$$ $$1$$ $$e\left(\frac{39}{86}\right)$$ $$e\left(\frac{21}{86}\right)$$ $$e\left(\frac{39}{43}\right)$$ $$e\left(\frac{59}{86}\right)$$ $$e\left(\frac{30}{43}\right)$$ $$e\left(\frac{7}{86}\right)$$ $$e\left(\frac{31}{86}\right)$$ $$e\left(\frac{21}{43}\right)$$ $$e\left(\frac{6}{43}\right)$$ $$e\left(\frac{37}{86}\right)$$
$$\chi_{173}(26,\cdot)$$ 173.f 172 yes $$-1$$ $$1$$ $$e\left(\frac{131}{172}\right)$$ $$e\left(\frac{97}{172}\right)$$ $$e\left(\frac{45}{86}\right)$$ $$e\left(\frac{121}{172}\right)$$ $$e\left(\frac{14}{43}\right)$$ $$e\left(\frac{61}{172}\right)$$ $$e\left(\frac{49}{172}\right)$$ $$e\left(\frac{11}{86}\right)$$ $$e\left(\frac{20}{43}\right)$$ $$e\left(\frac{89}{172}\right)$$
$$\chi_{173}(27,\cdot)$$ 173.f 172 yes $$-1$$ $$1$$ $$e\left(\frac{81}{172}\right)$$ $$e\left(\frac{123}{172}\right)$$ $$e\left(\frac{81}{86}\right)$$ $$e\left(\frac{63}{172}\right)$$ $$e\left(\frac{8}{43}\right)$$ $$e\left(\frac{127}{172}\right)$$ $$e\left(\frac{71}{172}\right)$$ $$e\left(\frac{37}{86}\right)$$ $$e\left(\frac{36}{43}\right)$$ $$e\left(\frac{143}{172}\right)$$
$$\chi_{173}(28,\cdot)$$ 173.f 172 yes $$-1$$ $$1$$ $$e\left(\frac{97}{172}\right)$$ $$e\left(\frac{39}{172}\right)$$ $$e\left(\frac{11}{86}\right)$$ $$e\left(\frac{171}{172}\right)$$ $$e\left(\frac{34}{43}\right)$$ $$e\left(\frac{99}{172}\right)$$ $$e\left(\frac{119}{172}\right)$$ $$e\left(\frac{39}{86}\right)$$ $$e\left(\frac{24}{43}\right)$$ $$e\left(\frac{167}{172}\right)$$
$$\chi_{173}(29,\cdot)$$ 173.d 43 yes $$1$$ $$1$$ $$e\left(\frac{36}{43}\right)$$ $$e\left(\frac{26}{43}\right)$$ $$e\left(\frac{29}{43}\right)$$ $$e\left(\frac{28}{43}\right)$$ $$e\left(\frac{19}{43}\right)$$ $$e\left(\frac{23}{43}\right)$$ $$e\left(\frac{22}{43}\right)$$ $$e\left(\frac{9}{43}\right)$$ $$e\left(\frac{21}{43}\right)$$ $$e\left(\frac{11}{43}\right)$$
$$\chi_{173}(30,\cdot)$$ 173.f 172 yes $$-1$$ $$1$$ $$e\left(\frac{67}{172}\right)$$ $$e\left(\frac{89}{172}\right)$$ $$e\left(\frac{67}{86}\right)$$ $$e\left(\frac{33}{172}\right)$$ $$e\left(\frac{39}{43}\right)$$ $$e\left(\frac{1}{172}\right)$$ $$e\left(\frac{29}{172}\right)$$ $$e\left(\frac{3}{86}\right)$$ $$e\left(\frac{25}{43}\right)$$ $$e\left(\frac{165}{172}\right)$$
$$\chi_{173}(31,\cdot)$$ 173.e 86 yes $$1$$ $$1$$ $$e\left(\frac{51}{86}\right)$$ $$e\left(\frac{1}{86}\right)$$ $$e\left(\frac{8}{43}\right)$$ $$e\left(\frac{11}{86}\right)$$ $$e\left(\frac{26}{43}\right)$$ $$e\left(\frac{29}{86}\right)$$ $$e\left(\frac{67}{86}\right)$$ $$e\left(\frac{1}{43}\right)$$ $$e\left(\frac{31}{43}\right)$$ $$e\left(\frac{55}{86}\right)$$
$$\chi_{173}(32,\cdot)$$ 173.f 172 yes $$-1$$ $$1$$ $$e\left(\frac{5}{172}\right)$$ $$e\left(\frac{135}{172}\right)$$ $$e\left(\frac{5}{86}\right)$$ $$e\left(\frac{23}{172}\right)$$ $$e\left(\frac{35}{43}\right)$$ $$e\left(\frac{131}{172}\right)$$ $$e\left(\frac{15}{172}\right)$$ $$e\left(\frac{49}{86}\right)$$ $$e\left(\frac{7}{43}\right)$$ $$e\left(\frac{115}{172}\right)$$