# Properties

 Modulus $643$ Structure $$C_{642}$$ Order $642$

Show commands: Pari/GP / SageMath

sage: H = DirichletGroup(643)

pari: g = idealstar(,643,2)

## Character group

 sage: G.order()  pari: g.no Order = 642 sage: H.invariants()  pari: g.cyc Structure = $$C_{642}$$ sage: H.gens()  pari: g.gen Generators = $\chi_{643}(11,\cdot)$

## First 32 of 642 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 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$
$$\chi_{643}(1,\cdot)$$ 643.a 1 no $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$ $$1$$
$$\chi_{643}(2,\cdot)$$ 643.f 214 yes $$-1$$ $$1$$ $$e\left(\frac{137}{214}\right)$$ $$e\left(\frac{7}{214}\right)$$ $$e\left(\frac{30}{107}\right)$$ $$e\left(\frac{11}{214}\right)$$ $$e\left(\frac{72}{107}\right)$$ $$e\left(\frac{104}{107}\right)$$ $$e\left(\frac{197}{214}\right)$$ $$e\left(\frac{7}{107}\right)$$ $$e\left(\frac{74}{107}\right)$$ $$e\left(\frac{151}{214}\right)$$
$$\chi_{643}(3,\cdot)$$ 643.f 214 yes $$-1$$ $$1$$ $$e\left(\frac{7}{214}\right)$$ $$e\left(\frac{155}{214}\right)$$ $$e\left(\frac{7}{107}\right)$$ $$e\left(\frac{213}{214}\right)$$ $$e\left(\frac{81}{107}\right)$$ $$e\left(\frac{10}{107}\right)$$ $$e\left(\frac{21}{214}\right)$$ $$e\left(\frac{48}{107}\right)$$ $$e\left(\frac{3}{107}\right)$$ $$e\left(\frac{103}{214}\right)$$
$$\chi_{643}(4,\cdot)$$ 643.e 107 yes $$1$$ $$1$$ $$e\left(\frac{30}{107}\right)$$ $$e\left(\frac{7}{107}\right)$$ $$e\left(\frac{60}{107}\right)$$ $$e\left(\frac{11}{107}\right)$$ $$e\left(\frac{37}{107}\right)$$ $$e\left(\frac{101}{107}\right)$$ $$e\left(\frac{90}{107}\right)$$ $$e\left(\frac{14}{107}\right)$$ $$e\left(\frac{41}{107}\right)$$ $$e\left(\frac{44}{107}\right)$$
$$\chi_{643}(5,\cdot)$$ 643.f 214 yes $$-1$$ $$1$$ $$e\left(\frac{11}{214}\right)$$ $$e\left(\frac{213}{214}\right)$$ $$e\left(\frac{11}{107}\right)$$ $$e\left(\frac{29}{214}\right)$$ $$e\left(\frac{5}{107}\right)$$ $$e\left(\frac{31}{107}\right)$$ $$e\left(\frac{33}{214}\right)$$ $$e\left(\frac{106}{107}\right)$$ $$e\left(\frac{20}{107}\right)$$ $$e\left(\frac{9}{214}\right)$$
$$\chi_{643}(6,\cdot)$$ 643.e 107 yes $$1$$ $$1$$ $$e\left(\frac{72}{107}\right)$$ $$e\left(\frac{81}{107}\right)$$ $$e\left(\frac{37}{107}\right)$$ $$e\left(\frac{5}{107}\right)$$ $$e\left(\frac{46}{107}\right)$$ $$e\left(\frac{7}{107}\right)$$ $$e\left(\frac{2}{107}\right)$$ $$e\left(\frac{55}{107}\right)$$ $$e\left(\frac{77}{107}\right)$$ $$e\left(\frac{20}{107}\right)$$
$$\chi_{643}(7,\cdot)$$ 643.g 321 yes $$1$$ $$1$$ $$e\left(\frac{104}{107}\right)$$ $$e\left(\frac{10}{107}\right)$$ $$e\left(\frac{101}{107}\right)$$ $$e\left(\frac{31}{107}\right)$$ $$e\left(\frac{7}{107}\right)$$ $$e\left(\frac{173}{321}\right)$$ $$e\left(\frac{98}{107}\right)$$ $$e\left(\frac{20}{107}\right)$$ $$e\left(\frac{28}{107}\right)$$ $$e\left(\frac{265}{321}\right)$$
$$\chi_{643}(8,\cdot)$$ 643.f 214 yes $$-1$$ $$1$$ $$e\left(\frac{197}{214}\right)$$ $$e\left(\frac{21}{214}\right)$$ $$e\left(\frac{90}{107}\right)$$ $$e\left(\frac{33}{214}\right)$$ $$e\left(\frac{2}{107}\right)$$ $$e\left(\frac{98}{107}\right)$$ $$e\left(\frac{163}{214}\right)$$ $$e\left(\frac{21}{107}\right)$$ $$e\left(\frac{8}{107}\right)$$ $$e\left(\frac{25}{214}\right)$$
$$\chi_{643}(9,\cdot)$$ 643.e 107 yes $$1$$ $$1$$ $$e\left(\frac{7}{107}\right)$$ $$e\left(\frac{48}{107}\right)$$ $$e\left(\frac{14}{107}\right)$$ $$e\left(\frac{106}{107}\right)$$ $$e\left(\frac{55}{107}\right)$$ $$e\left(\frac{20}{107}\right)$$ $$e\left(\frac{21}{107}\right)$$ $$e\left(\frac{96}{107}\right)$$ $$e\left(\frac{6}{107}\right)$$ $$e\left(\frac{103}{107}\right)$$
$$\chi_{643}(10,\cdot)$$ 643.e 107 yes $$1$$ $$1$$ $$e\left(\frac{74}{107}\right)$$ $$e\left(\frac{3}{107}\right)$$ $$e\left(\frac{41}{107}\right)$$ $$e\left(\frac{20}{107}\right)$$ $$e\left(\frac{77}{107}\right)$$ $$e\left(\frac{28}{107}\right)$$ $$e\left(\frac{8}{107}\right)$$ $$e\left(\frac{6}{107}\right)$$ $$e\left(\frac{94}{107}\right)$$ $$e\left(\frac{80}{107}\right)$$
$$\chi_{643}(11,\cdot)$$ 643.h 642 yes $$-1$$ $$1$$ $$e\left(\frac{151}{214}\right)$$ $$e\left(\frac{103}{214}\right)$$ $$e\left(\frac{44}{107}\right)$$ $$e\left(\frac{9}{214}\right)$$ $$e\left(\frac{20}{107}\right)$$ $$e\left(\frac{265}{321}\right)$$ $$e\left(\frac{25}{214}\right)$$ $$e\left(\frac{103}{107}\right)$$ $$e\left(\frac{80}{107}\right)$$ $$e\left(\frac{1}{642}\right)$$
$$\chi_{643}(12,\cdot)$$ 643.f 214 yes $$-1$$ $$1$$ $$e\left(\frac{67}{214}\right)$$ $$e\left(\frac{169}{214}\right)$$ $$e\left(\frac{67}{107}\right)$$ $$e\left(\frac{21}{214}\right)$$ $$e\left(\frac{11}{107}\right)$$ $$e\left(\frac{4}{107}\right)$$ $$e\left(\frac{201}{214}\right)$$ $$e\left(\frac{62}{107}\right)$$ $$e\left(\frac{44}{107}\right)$$ $$e\left(\frac{191}{214}\right)$$
$$\chi_{643}(13,\cdot)$$ 643.h 642 yes $$-1$$ $$1$$ $$e\left(\frac{49}{214}\right)$$ $$e\left(\frac{15}{214}\right)$$ $$e\left(\frac{49}{107}\right)$$ $$e\left(\frac{207}{214}\right)$$ $$e\left(\frac{32}{107}\right)$$ $$e\left(\frac{317}{321}\right)$$ $$e\left(\frac{147}{214}\right)$$ $$e\left(\frac{15}{107}\right)$$ $$e\left(\frac{21}{107}\right)$$ $$e\left(\frac{23}{642}\right)$$
$$\chi_{643}(14,\cdot)$$ 643.h 642 yes $$-1$$ $$1$$ $$e\left(\frac{131}{214}\right)$$ $$e\left(\frac{27}{214}\right)$$ $$e\left(\frac{24}{107}\right)$$ $$e\left(\frac{73}{214}\right)$$ $$e\left(\frac{79}{107}\right)$$ $$e\left(\frac{164}{321}\right)$$ $$e\left(\frac{179}{214}\right)$$ $$e\left(\frac{27}{107}\right)$$ $$e\left(\frac{102}{107}\right)$$ $$e\left(\frac{341}{642}\right)$$
$$\chi_{643}(15,\cdot)$$ 643.e 107 yes $$1$$ $$1$$ $$e\left(\frac{9}{107}\right)$$ $$e\left(\frac{77}{107}\right)$$ $$e\left(\frac{18}{107}\right)$$ $$e\left(\frac{14}{107}\right)$$ $$e\left(\frac{86}{107}\right)$$ $$e\left(\frac{41}{107}\right)$$ $$e\left(\frac{27}{107}\right)$$ $$e\left(\frac{47}{107}\right)$$ $$e\left(\frac{23}{107}\right)$$ $$e\left(\frac{56}{107}\right)$$
$$\chi_{643}(16,\cdot)$$ 643.e 107 yes $$1$$ $$1$$ $$e\left(\frac{60}{107}\right)$$ $$e\left(\frac{14}{107}\right)$$ $$e\left(\frac{13}{107}\right)$$ $$e\left(\frac{22}{107}\right)$$ $$e\left(\frac{74}{107}\right)$$ $$e\left(\frac{95}{107}\right)$$ $$e\left(\frac{73}{107}\right)$$ $$e\left(\frac{28}{107}\right)$$ $$e\left(\frac{82}{107}\right)$$ $$e\left(\frac{88}{107}\right)$$
$$\chi_{643}(17,\cdot)$$ 643.h 642 yes $$-1$$ $$1$$ $$e\left(\frac{15}{214}\right)$$ $$e\left(\frac{57}{214}\right)$$ $$e\left(\frac{15}{107}\right)$$ $$e\left(\frac{59}{214}\right)$$ $$e\left(\frac{36}{107}\right)$$ $$e\left(\frac{49}{321}\right)$$ $$e\left(\frac{45}{214}\right)$$ $$e\left(\frac{57}{107}\right)$$ $$e\left(\frac{37}{107}\right)$$ $$e\left(\frac{601}{642}\right)$$
$$\chi_{643}(18,\cdot)$$ 643.f 214 yes $$-1$$ $$1$$ $$e\left(\frac{151}{214}\right)$$ $$e\left(\frac{103}{214}\right)$$ $$e\left(\frac{44}{107}\right)$$ $$e\left(\frac{9}{214}\right)$$ $$e\left(\frac{20}{107}\right)$$ $$e\left(\frac{17}{107}\right)$$ $$e\left(\frac{25}{214}\right)$$ $$e\left(\frac{103}{107}\right)$$ $$e\left(\frac{80}{107}\right)$$ $$e\left(\frac{143}{214}\right)$$
$$\chi_{643}(19,\cdot)$$ 643.h 642 yes $$-1$$ $$1$$ $$e\left(\frac{69}{214}\right)$$ $$e\left(\frac{91}{214}\right)$$ $$e\left(\frac{69}{107}\right)$$ $$e\left(\frac{143}{214}\right)$$ $$e\left(\frac{80}{107}\right)$$ $$e\left(\frac{311}{321}\right)$$ $$e\left(\frac{207}{214}\right)$$ $$e\left(\frac{91}{107}\right)$$ $$e\left(\frac{106}{107}\right)$$ $$e\left(\frac{539}{642}\right)$$
$$\chi_{643}(20,\cdot)$$ 643.f 214 yes $$-1$$ $$1$$ $$e\left(\frac{71}{214}\right)$$ $$e\left(\frac{13}{214}\right)$$ $$e\left(\frac{71}{107}\right)$$ $$e\left(\frac{51}{214}\right)$$ $$e\left(\frac{42}{107}\right)$$ $$e\left(\frac{25}{107}\right)$$ $$e\left(\frac{213}{214}\right)$$ $$e\left(\frac{13}{107}\right)$$ $$e\left(\frac{61}{107}\right)$$ $$e\left(\frac{97}{214}\right)$$
$$\chi_{643}(21,\cdot)$$ 643.h 642 yes $$-1$$ $$1$$ $$e\left(\frac{1}{214}\right)$$ $$e\left(\frac{175}{214}\right)$$ $$e\left(\frac{1}{107}\right)$$ $$e\left(\frac{61}{214}\right)$$ $$e\left(\frac{88}{107}\right)$$ $$e\left(\frac{203}{321}\right)$$ $$e\left(\frac{3}{214}\right)$$ $$e\left(\frac{68}{107}\right)$$ $$e\left(\frac{31}{107}\right)$$ $$e\left(\frac{197}{642}\right)$$
$$\chi_{643}(22,\cdot)$$ 643.g 321 yes $$1$$ $$1$$ $$e\left(\frac{37}{107}\right)$$ $$e\left(\frac{55}{107}\right)$$ $$e\left(\frac{74}{107}\right)$$ $$e\left(\frac{10}{107}\right)$$ $$e\left(\frac{92}{107}\right)$$ $$e\left(\frac{256}{321}\right)$$ $$e\left(\frac{4}{107}\right)$$ $$e\left(\frac{3}{107}\right)$$ $$e\left(\frac{47}{107}\right)$$ $$e\left(\frac{227}{321}\right)$$
$$\chi_{643}(23,\cdot)$$ 643.g 321 yes $$1$$ $$1$$ $$e\left(\frac{80}{107}\right)$$ $$e\left(\frac{90}{107}\right)$$ $$e\left(\frac{53}{107}\right)$$ $$e\left(\frac{65}{107}\right)$$ $$e\left(\frac{63}{107}\right)$$ $$e\left(\frac{166}{321}\right)$$ $$e\left(\frac{26}{107}\right)$$ $$e\left(\frac{73}{107}\right)$$ $$e\left(\frac{38}{107}\right)$$ $$e\left(\frac{245}{321}\right)$$
$$\chi_{643}(24,\cdot)$$ 643.e 107 yes $$1$$ $$1$$ $$e\left(\frac{102}{107}\right)$$ $$e\left(\frac{88}{107}\right)$$ $$e\left(\frac{97}{107}\right)$$ $$e\left(\frac{16}{107}\right)$$ $$e\left(\frac{83}{107}\right)$$ $$e\left(\frac{1}{107}\right)$$ $$e\left(\frac{92}{107}\right)$$ $$e\left(\frac{69}{107}\right)$$ $$e\left(\frac{11}{107}\right)$$ $$e\left(\frac{64}{107}\right)$$
$$\chi_{643}(25,\cdot)$$ 643.e 107 yes $$1$$ $$1$$ $$e\left(\frac{11}{107}\right)$$ $$e\left(\frac{106}{107}\right)$$ $$e\left(\frac{22}{107}\right)$$ $$e\left(\frac{29}{107}\right)$$ $$e\left(\frac{10}{107}\right)$$ $$e\left(\frac{62}{107}\right)$$ $$e\left(\frac{33}{107}\right)$$ $$e\left(\frac{105}{107}\right)$$ $$e\left(\frac{40}{107}\right)$$ $$e\left(\frac{9}{107}\right)$$
$$\chi_{643}(26,\cdot)$$ 643.g 321 yes $$1$$ $$1$$ $$e\left(\frac{93}{107}\right)$$ $$e\left(\frac{11}{107}\right)$$ $$e\left(\frac{79}{107}\right)$$ $$e\left(\frac{2}{107}\right)$$ $$e\left(\frac{104}{107}\right)$$ $$e\left(\frac{308}{321}\right)$$ $$e\left(\frac{65}{107}\right)$$ $$e\left(\frac{22}{107}\right)$$ $$e\left(\frac{95}{107}\right)$$ $$e\left(\frac{238}{321}\right)$$
$$\chi_{643}(27,\cdot)$$ 643.f 214 yes $$-1$$ $$1$$ $$e\left(\frac{21}{214}\right)$$ $$e\left(\frac{37}{214}\right)$$ $$e\left(\frac{21}{107}\right)$$ $$e\left(\frac{211}{214}\right)$$ $$e\left(\frac{29}{107}\right)$$ $$e\left(\frac{30}{107}\right)$$ $$e\left(\frac{63}{214}\right)$$ $$e\left(\frac{37}{107}\right)$$ $$e\left(\frac{9}{107}\right)$$ $$e\left(\frac{95}{214}\right)$$
$$\chi_{643}(28,\cdot)$$ 643.g 321 yes $$1$$ $$1$$ $$e\left(\frac{27}{107}\right)$$ $$e\left(\frac{17}{107}\right)$$ $$e\left(\frac{54}{107}\right)$$ $$e\left(\frac{42}{107}\right)$$ $$e\left(\frac{44}{107}\right)$$ $$e\left(\frac{155}{321}\right)$$ $$e\left(\frac{81}{107}\right)$$ $$e\left(\frac{34}{107}\right)$$ $$e\left(\frac{69}{107}\right)$$ $$e\left(\frac{76}{321}\right)$$
$$\chi_{643}(29,\cdot)$$ 643.g 321 yes $$1$$ $$1$$ $$e\left(\frac{22}{107}\right)$$ $$e\left(\frac{105}{107}\right)$$ $$e\left(\frac{44}{107}\right)$$ $$e\left(\frac{58}{107}\right)$$ $$e\left(\frac{20}{107}\right)$$ $$e\left(\frac{158}{321}\right)$$ $$e\left(\frac{66}{107}\right)$$ $$e\left(\frac{103}{107}\right)$$ $$e\left(\frac{80}{107}\right)$$ $$e\left(\frac{268}{321}\right)$$
$$\chi_{643}(30,\cdot)$$ 643.f 214 yes $$-1$$ $$1$$ $$e\left(\frac{155}{214}\right)$$ $$e\left(\frac{161}{214}\right)$$ $$e\left(\frac{48}{107}\right)$$ $$e\left(\frac{39}{214}\right)$$ $$e\left(\frac{51}{107}\right)$$ $$e\left(\frac{38}{107}\right)$$ $$e\left(\frac{37}{214}\right)$$ $$e\left(\frac{54}{107}\right)$$ $$e\left(\frac{97}{107}\right)$$ $$e\left(\frac{49}{214}\right)$$
$$\chi_{643}(31,\cdot)$$ 643.g 321 yes $$1$$ $$1$$ $$e\left(\frac{98}{107}\right)$$ $$e\left(\frac{30}{107}\right)$$ $$e\left(\frac{89}{107}\right)$$ $$e\left(\frac{93}{107}\right)$$ $$e\left(\frac{21}{107}\right)$$ $$e\left(\frac{91}{321}\right)$$ $$e\left(\frac{80}{107}\right)$$ $$e\left(\frac{60}{107}\right)$$ $$e\left(\frac{84}{107}\right)$$ $$e\left(\frac{260}{321}\right)$$
$$\chi_{643}(32,\cdot)$$ 643.f 214 yes $$-1$$ $$1$$ $$e\left(\frac{43}{214}\right)$$ $$e\left(\frac{35}{214}\right)$$ $$e\left(\frac{43}{107}\right)$$ $$e\left(\frac{55}{214}\right)$$ $$e\left(\frac{39}{107}\right)$$ $$e\left(\frac{92}{107}\right)$$ $$e\left(\frac{129}{214}\right)$$ $$e\left(\frac{35}{107}\right)$$ $$e\left(\frac{49}{107}\right)$$ $$e\left(\frac{113}{214}\right)$$