# Properties

 Label 643.f Modulus $643$ Conductor $643$ Order $214$ Real no Primitive yes Minimal yes Parity odd

# Related objects

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter

sage: H = DirichletGroup(643, base_ring=CyclotomicField(214))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([151]))

sage: chi.galois_orbit()

pari: [g,chi] = znchar(Mod(2,643))

pari: order = charorder(g,chi)

pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]

## Basic properties

 Modulus: $$643$$ Conductor: $$643$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$214$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: yes sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Related number fields

 Field of values: $\Q(\zeta_{107})$ Fixed field: Number field defined by a degree 214 polynomial (not computed)

## First 31 of 106 characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$
$$\chi_{643}(2,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{137}{214}\right)$$ $$e\left(\frac{7}{214}\right)$$ $$e\left(\frac{30}{107}\right)$$ $$e\left(\frac{11}{214}\right)$$ $$e\left(\frac{72}{107}\right)$$ $$e\left(\frac{104}{107}\right)$$ $$e\left(\frac{197}{214}\right)$$ $$e\left(\frac{7}{107}\right)$$ $$e\left(\frac{74}{107}\right)$$ $$e\left(\frac{151}{214}\right)$$
$$\chi_{643}(3,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{7}{214}\right)$$ $$e\left(\frac{155}{214}\right)$$ $$e\left(\frac{7}{107}\right)$$ $$e\left(\frac{213}{214}\right)$$ $$e\left(\frac{81}{107}\right)$$ $$e\left(\frac{10}{107}\right)$$ $$e\left(\frac{21}{214}\right)$$ $$e\left(\frac{48}{107}\right)$$ $$e\left(\frac{3}{107}\right)$$ $$e\left(\frac{103}{214}\right)$$
$$\chi_{643}(5,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{11}{214}\right)$$ $$e\left(\frac{213}{214}\right)$$ $$e\left(\frac{11}{107}\right)$$ $$e\left(\frac{29}{214}\right)$$ $$e\left(\frac{5}{107}\right)$$ $$e\left(\frac{31}{107}\right)$$ $$e\left(\frac{33}{214}\right)$$ $$e\left(\frac{106}{107}\right)$$ $$e\left(\frac{20}{107}\right)$$ $$e\left(\frac{9}{214}\right)$$
$$\chi_{643}(8,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{197}{214}\right)$$ $$e\left(\frac{21}{214}\right)$$ $$e\left(\frac{90}{107}\right)$$ $$e\left(\frac{33}{214}\right)$$ $$e\left(\frac{2}{107}\right)$$ $$e\left(\frac{98}{107}\right)$$ $$e\left(\frac{163}{214}\right)$$ $$e\left(\frac{21}{107}\right)$$ $$e\left(\frac{8}{107}\right)$$ $$e\left(\frac{25}{214}\right)$$
$$\chi_{643}(12,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{67}{214}\right)$$ $$e\left(\frac{169}{214}\right)$$ $$e\left(\frac{67}{107}\right)$$ $$e\left(\frac{21}{214}\right)$$ $$e\left(\frac{11}{107}\right)$$ $$e\left(\frac{4}{107}\right)$$ $$e\left(\frac{201}{214}\right)$$ $$e\left(\frac{62}{107}\right)$$ $$e\left(\frac{44}{107}\right)$$ $$e\left(\frac{191}{214}\right)$$
$$\chi_{643}(18,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{151}{214}\right)$$ $$e\left(\frac{103}{214}\right)$$ $$e\left(\frac{44}{107}\right)$$ $$e\left(\frac{9}{214}\right)$$ $$e\left(\frac{20}{107}\right)$$ $$e\left(\frac{17}{107}\right)$$ $$e\left(\frac{25}{214}\right)$$ $$e\left(\frac{103}{107}\right)$$ $$e\left(\frac{80}{107}\right)$$ $$e\left(\frac{143}{214}\right)$$
$$\chi_{643}(20,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{71}{214}\right)$$ $$e\left(\frac{13}{214}\right)$$ $$e\left(\frac{71}{107}\right)$$ $$e\left(\frac{51}{214}\right)$$ $$e\left(\frac{42}{107}\right)$$ $$e\left(\frac{25}{107}\right)$$ $$e\left(\frac{213}{214}\right)$$ $$e\left(\frac{13}{107}\right)$$ $$e\left(\frac{61}{107}\right)$$ $$e\left(\frac{97}{214}\right)$$
$$\chi_{643}(27,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{21}{214}\right)$$ $$e\left(\frac{37}{214}\right)$$ $$e\left(\frac{21}{107}\right)$$ $$e\left(\frac{211}{214}\right)$$ $$e\left(\frac{29}{107}\right)$$ $$e\left(\frac{30}{107}\right)$$ $$e\left(\frac{63}{214}\right)$$ $$e\left(\frac{37}{107}\right)$$ $$e\left(\frac{9}{107}\right)$$ $$e\left(\frac{95}{214}\right)$$
$$\chi_{643}(30,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{155}{214}\right)$$ $$e\left(\frac{161}{214}\right)$$ $$e\left(\frac{48}{107}\right)$$ $$e\left(\frac{39}{214}\right)$$ $$e\left(\frac{51}{107}\right)$$ $$e\left(\frac{38}{107}\right)$$ $$e\left(\frac{37}{214}\right)$$ $$e\left(\frac{54}{107}\right)$$ $$e\left(\frac{97}{107}\right)$$ $$e\left(\frac{49}{214}\right)$$
$$\chi_{643}(32,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{43}{214}\right)$$ $$e\left(\frac{35}{214}\right)$$ $$e\left(\frac{43}{107}\right)$$ $$e\left(\frac{55}{214}\right)$$ $$e\left(\frac{39}{107}\right)$$ $$e\left(\frac{92}{107}\right)$$ $$e\left(\frac{129}{214}\right)$$ $$e\left(\frac{35}{107}\right)$$ $$e\left(\frac{49}{107}\right)$$ $$e\left(\frac{113}{214}\right)$$
$$\chi_{643}(43,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{119}{214}\right)$$ $$e\left(\frac{67}{214}\right)$$ $$e\left(\frac{12}{107}\right)$$ $$e\left(\frac{197}{214}\right)$$ $$e\left(\frac{93}{107}\right)$$ $$e\left(\frac{63}{107}\right)$$ $$e\left(\frac{143}{214}\right)$$ $$e\left(\frac{67}{107}\right)$$ $$e\left(\frac{51}{107}\right)$$ $$e\left(\frac{39}{214}\right)$$
$$\chi_{643}(45,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{25}{214}\right)$$ $$e\left(\frac{95}{214}\right)$$ $$e\left(\frac{25}{107}\right)$$ $$e\left(\frac{27}{214}\right)$$ $$e\left(\frac{60}{107}\right)$$ $$e\left(\frac{51}{107}\right)$$ $$e\left(\frac{75}{214}\right)$$ $$e\left(\frac{95}{107}\right)$$ $$e\left(\frac{26}{107}\right)$$ $$e\left(\frac{1}{214}\right)$$
$$\chi_{643}(48,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{127}{214}\right)$$ $$e\left(\frac{183}{214}\right)$$ $$e\left(\frac{20}{107}\right)$$ $$e\left(\frac{43}{214}\right)$$ $$e\left(\frac{48}{107}\right)$$ $$e\left(\frac{105}{107}\right)$$ $$e\left(\frac{167}{214}\right)$$ $$e\left(\frac{76}{107}\right)$$ $$e\left(\frac{85}{107}\right)$$ $$e\left(\frac{65}{214}\right)$$
$$\chi_{643}(50,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{159}{214}\right)$$ $$e\left(\frac{5}{214}\right)$$ $$e\left(\frac{52}{107}\right)$$ $$e\left(\frac{69}{214}\right)$$ $$e\left(\frac{82}{107}\right)$$ $$e\left(\frac{59}{107}\right)$$ $$e\left(\frac{49}{214}\right)$$ $$e\left(\frac{5}{107}\right)$$ $$e\left(\frac{7}{107}\right)$$ $$e\left(\frac{169}{214}\right)$$
$$\chi_{643}(67,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{87}{214}\right)$$ $$e\left(\frac{31}{214}\right)$$ $$e\left(\frac{87}{107}\right)$$ $$e\left(\frac{171}{214}\right)$$ $$e\left(\frac{59}{107}\right)$$ $$e\left(\frac{2}{107}\right)$$ $$e\left(\frac{47}{214}\right)$$ $$e\left(\frac{31}{107}\right)$$ $$e\left(\frac{22}{107}\right)$$ $$e\left(\frac{149}{214}\right)$$
$$\chi_{643}(71,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{153}{214}\right)$$ $$e\left(\frac{25}{214}\right)$$ $$e\left(\frac{46}{107}\right)$$ $$e\left(\frac{131}{214}\right)$$ $$e\left(\frac{89}{107}\right)$$ $$e\left(\frac{81}{107}\right)$$ $$e\left(\frac{31}{214}\right)$$ $$e\left(\frac{25}{107}\right)$$ $$e\left(\frac{35}{107}\right)$$ $$e\left(\frac{203}{214}\right)$$
$$\chi_{643}(72,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{211}{214}\right)$$ $$e\left(\frac{117}{214}\right)$$ $$e\left(\frac{104}{107}\right)$$ $$e\left(\frac{31}{214}\right)$$ $$e\left(\frac{57}{107}\right)$$ $$e\left(\frac{11}{107}\right)$$ $$e\left(\frac{205}{214}\right)$$ $$e\left(\frac{10}{107}\right)$$ $$e\left(\frac{14}{107}\right)$$ $$e\left(\frac{17}{214}\right)$$
$$\chi_{643}(75,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{29}{214}\right)$$ $$e\left(\frac{153}{214}\right)$$ $$e\left(\frac{29}{107}\right)$$ $$e\left(\frac{57}{214}\right)$$ $$e\left(\frac{91}{107}\right)$$ $$e\left(\frac{72}{107}\right)$$ $$e\left(\frac{87}{214}\right)$$ $$e\left(\frac{46}{107}\right)$$ $$e\left(\frac{43}{107}\right)$$ $$e\left(\frac{121}{214}\right)$$
$$\chi_{643}(77,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{145}{214}\right)$$ $$e\left(\frac{123}{214}\right)$$ $$e\left(\frac{38}{107}\right)$$ $$e\left(\frac{71}{214}\right)$$ $$e\left(\frac{27}{107}\right)$$ $$e\left(\frac{39}{107}\right)$$ $$e\left(\frac{7}{214}\right)$$ $$e\left(\frac{16}{107}\right)$$ $$e\left(\frac{1}{107}\right)$$ $$e\left(\frac{177}{214}\right)$$
$$\chi_{643}(80,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{131}{214}\right)$$ $$e\left(\frac{27}{214}\right)$$ $$e\left(\frac{24}{107}\right)$$ $$e\left(\frac{73}{214}\right)$$ $$e\left(\frac{79}{107}\right)$$ $$e\left(\frac{19}{107}\right)$$ $$e\left(\frac{179}{214}\right)$$ $$e\left(\frac{27}{107}\right)$$ $$e\left(\frac{102}{107}\right)$$ $$e\left(\frac{185}{214}\right)$$
$$\chi_{643}(103,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{199}{214}\right)$$ $$e\left(\frac{157}{214}\right)$$ $$e\left(\frac{92}{107}\right)$$ $$e\left(\frac{155}{214}\right)$$ $$e\left(\frac{71}{107}\right)$$ $$e\left(\frac{55}{107}\right)$$ $$e\left(\frac{169}{214}\right)$$ $$e\left(\frac{50}{107}\right)$$ $$e\left(\frac{70}{107}\right)$$ $$e\left(\frac{85}{214}\right)$$
$$\chi_{643}(107,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{177}{214}\right)$$ $$e\left(\frac{159}{214}\right)$$ $$e\left(\frac{70}{107}\right)$$ $$e\left(\frac{97}{214}\right)$$ $$e\left(\frac{61}{107}\right)$$ $$e\left(\frac{100}{107}\right)$$ $$e\left(\frac{103}{214}\right)$$ $$e\left(\frac{52}{107}\right)$$ $$e\left(\frac{30}{107}\right)$$ $$e\left(\frac{67}{214}\right)$$
$$\chi_{643}(108,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{81}{214}\right)$$ $$e\left(\frac{51}{214}\right)$$ $$e\left(\frac{81}{107}\right)$$ $$e\left(\frac{19}{214}\right)$$ $$e\left(\frac{66}{107}\right)$$ $$e\left(\frac{24}{107}\right)$$ $$e\left(\frac{29}{214}\right)$$ $$e\left(\frac{51}{107}\right)$$ $$e\left(\frac{50}{107}\right)$$ $$e\left(\frac{183}{214}\right)$$
$$\chi_{643}(119,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{9}{214}\right)$$ $$e\left(\frac{77}{214}\right)$$ $$e\left(\frac{9}{107}\right)$$ $$e\left(\frac{121}{214}\right)$$ $$e\left(\frac{43}{107}\right)$$ $$e\left(\frac{74}{107}\right)$$ $$e\left(\frac{27}{214}\right)$$ $$e\left(\frac{77}{107}\right)$$ $$e\left(\frac{65}{107}\right)$$ $$e\left(\frac{163}{214}\right)$$
$$\chi_{643}(120,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{214}\right)$$ $$e\left(\frac{175}{214}\right)$$ $$e\left(\frac{1}{107}\right)$$ $$e\left(\frac{61}{214}\right)$$ $$e\left(\frac{88}{107}\right)$$ $$e\left(\frac{32}{107}\right)$$ $$e\left(\frac{3}{214}\right)$$ $$e\left(\frac{68}{107}\right)$$ $$e\left(\frac{31}{107}\right)$$ $$e\left(\frac{137}{214}\right)$$
$$\chi_{643}(125,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{33}{214}\right)$$ $$e\left(\frac{211}{214}\right)$$ $$e\left(\frac{33}{107}\right)$$ $$e\left(\frac{87}{214}\right)$$ $$e\left(\frac{15}{107}\right)$$ $$e\left(\frac{93}{107}\right)$$ $$e\left(\frac{99}{214}\right)$$ $$e\left(\frac{104}{107}\right)$$ $$e\left(\frac{60}{107}\right)$$ $$e\left(\frac{27}{214}\right)$$
$$\chi_{643}(127,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{79}{214}\right)$$ $$e\left(\frac{129}{214}\right)$$ $$e\left(\frac{79}{107}\right)$$ $$e\left(\frac{111}{214}\right)$$ $$e\left(\frac{104}{107}\right)$$ $$e\left(\frac{67}{107}\right)$$ $$e\left(\frac{23}{214}\right)$$ $$e\left(\frac{22}{107}\right)$$ $$e\left(\frac{95}{107}\right)$$ $$e\left(\frac{123}{214}\right)$$
$$\chi_{643}(128,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{103}{214}\right)$$ $$e\left(\frac{49}{214}\right)$$ $$e\left(\frac{103}{107}\right)$$ $$e\left(\frac{77}{214}\right)$$ $$e\left(\frac{76}{107}\right)$$ $$e\left(\frac{86}{107}\right)$$ $$e\left(\frac{95}{214}\right)$$ $$e\left(\frac{49}{107}\right)$$ $$e\left(\frac{90}{107}\right)$$ $$e\left(\frac{201}{214}\right)$$
$$\chi_{643}(157,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{65}{214}\right)$$ $$e\left(\frac{33}{214}\right)$$ $$e\left(\frac{65}{107}\right)$$ $$e\left(\frac{113}{214}\right)$$ $$e\left(\frac{49}{107}\right)$$ $$e\left(\frac{47}{107}\right)$$ $$e\left(\frac{195}{214}\right)$$ $$e\left(\frac{33}{107}\right)$$ $$e\left(\frac{89}{107}\right)$$ $$e\left(\frac{131}{214}\right)$$
$$\chi_{643}(162,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{165}{214}\right)$$ $$e\left(\frac{199}{214}\right)$$ $$e\left(\frac{58}{107}\right)$$ $$e\left(\frac{7}{214}\right)$$ $$e\left(\frac{75}{107}\right)$$ $$e\left(\frac{37}{107}\right)$$ $$e\left(\frac{67}{214}\right)$$ $$e\left(\frac{92}{107}\right)$$ $$e\left(\frac{86}{107}\right)$$ $$e\left(\frac{135}{214}\right)$$
$$\chi_{643}(172,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{179}{214}\right)$$ $$e\left(\frac{81}{214}\right)$$ $$e\left(\frac{72}{107}\right)$$ $$e\left(\frac{5}{214}\right)$$ $$e\left(\frac{23}{107}\right)$$ $$e\left(\frac{57}{107}\right)$$ $$e\left(\frac{109}{214}\right)$$ $$e\left(\frac{81}{107}\right)$$ $$e\left(\frac{92}{107}\right)$$ $$e\left(\frac{127}{214}\right)$$