# Properties

 Label 173.d Modulus $173$ Conductor $173$ Order $43$ Real no Primitive yes Minimal yes Parity even

# Related objects

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

sage: H = DirichletGroup(173, base_ring=CyclotomicField(86))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([14]))

sage: chi.galois_orbit()

pari: [g,chi] = znchar(Mod(6,173))

pari: order = charorder(g,chi)

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

## Basic properties

 Modulus: $$173$$ Conductor: $$173$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$43$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: yes sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Related number fields

 Field of values: $\Q(\zeta_{43})$ Fixed field: 43.43.9952594992767664919302480397055915291864099597331865326873171797085952964991380209309055259529.1

## First 31 of 42 characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$
$$\chi_{173}(6,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{7}{43}\right)$$ $$e\left(\frac{17}{43}\right)$$ $$e\left(\frac{14}{43}\right)$$ $$e\left(\frac{15}{43}\right)$$ $$e\left(\frac{24}{43}\right)$$ $$e\left(\frac{20}{43}\right)$$ $$e\left(\frac{21}{43}\right)$$ $$e\left(\frac{34}{43}\right)$$ $$e\left(\frac{22}{43}\right)$$ $$e\left(\frac{32}{43}\right)$$
$$\chi_{173}(10,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{10}{43}\right)$$ $$e\left(\frac{12}{43}\right)$$ $$e\left(\frac{20}{43}\right)$$ $$e\left(\frac{3}{43}\right)$$ $$e\left(\frac{22}{43}\right)$$ $$e\left(\frac{4}{43}\right)$$ $$e\left(\frac{30}{43}\right)$$ $$e\left(\frac{24}{43}\right)$$ $$e\left(\frac{13}{43}\right)$$ $$e\left(\frac{15}{43}\right)$$
$$\chi_{173}(14,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{24}{43}\right)$$ $$e\left(\frac{3}{43}\right)$$ $$e\left(\frac{5}{43}\right)$$ $$e\left(\frac{33}{43}\right)$$ $$e\left(\frac{27}{43}\right)$$ $$e\left(\frac{1}{43}\right)$$ $$e\left(\frac{29}{43}\right)$$ $$e\left(\frac{6}{43}\right)$$ $$e\left(\frac{14}{43}\right)$$ $$e\left(\frac{36}{43}\right)$$
$$\chi_{173}(16,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{43}\right)$$ $$e\left(\frac{27}{43}\right)$$ $$e\left(\frac{2}{43}\right)$$ $$e\left(\frac{39}{43}\right)$$ $$e\left(\frac{28}{43}\right)$$ $$e\left(\frac{9}{43}\right)$$ $$e\left(\frac{3}{43}\right)$$ $$e\left(\frac{11}{43}\right)$$ $$e\left(\frac{40}{43}\right)$$ $$e\left(\frac{23}{43}\right)$$
$$\chi_{173}(22,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{6}{43}\right)$$ $$e\left(\frac{33}{43}\right)$$ $$e\left(\frac{12}{43}\right)$$ $$e\left(\frac{19}{43}\right)$$ $$e\left(\frac{39}{43}\right)$$ $$e\left(\frac{11}{43}\right)$$ $$e\left(\frac{18}{43}\right)$$ $$e\left(\frac{23}{43}\right)$$ $$e\left(\frac{25}{43}\right)$$ $$e\left(\frac{9}{43}\right)$$
$$\chi_{173}(23,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{5}{43}\right)$$ $$e\left(\frac{6}{43}\right)$$ $$e\left(\frac{10}{43}\right)$$ $$e\left(\frac{23}{43}\right)$$ $$e\left(\frac{11}{43}\right)$$ $$e\left(\frac{2}{43}\right)$$ $$e\left(\frac{15}{43}\right)$$ $$e\left(\frac{12}{43}\right)$$ $$e\left(\frac{28}{43}\right)$$ $$e\left(\frac{29}{43}\right)$$
$$\chi_{173}(29,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{36}{43}\right)$$ $$e\left(\frac{26}{43}\right)$$ $$e\left(\frac{29}{43}\right)$$ $$e\left(\frac{28}{43}\right)$$ $$e\left(\frac{19}{43}\right)$$ $$e\left(\frac{23}{43}\right)$$ $$e\left(\frac{22}{43}\right)$$ $$e\left(\frac{9}{43}\right)$$ $$e\left(\frac{21}{43}\right)$$ $$e\left(\frac{11}{43}\right)$$
$$\chi_{173}(36,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{14}{43}\right)$$ $$e\left(\frac{34}{43}\right)$$ $$e\left(\frac{28}{43}\right)$$ $$e\left(\frac{30}{43}\right)$$ $$e\left(\frac{5}{43}\right)$$ $$e\left(\frac{40}{43}\right)$$ $$e\left(\frac{42}{43}\right)$$ $$e\left(\frac{25}{43}\right)$$ $$e\left(\frac{1}{43}\right)$$ $$e\left(\frac{21}{43}\right)$$
$$\chi_{173}(43,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{21}{43}\right)$$ $$e\left(\frac{8}{43}\right)$$ $$e\left(\frac{42}{43}\right)$$ $$e\left(\frac{2}{43}\right)$$ $$e\left(\frac{29}{43}\right)$$ $$e\left(\frac{17}{43}\right)$$ $$e\left(\frac{20}{43}\right)$$ $$e\left(\frac{16}{43}\right)$$ $$e\left(\frac{23}{43}\right)$$ $$e\left(\frac{10}{43}\right)$$
$$\chi_{173}(47,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{16}{43}\right)$$ $$e\left(\frac{2}{43}\right)$$ $$e\left(\frac{32}{43}\right)$$ $$e\left(\frac{22}{43}\right)$$ $$e\left(\frac{18}{43}\right)$$ $$e\left(\frac{15}{43}\right)$$ $$e\left(\frac{5}{43}\right)$$ $$e\left(\frac{4}{43}\right)$$ $$e\left(\frac{38}{43}\right)$$ $$e\left(\frac{24}{43}\right)$$
$$\chi_{173}(51,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{25}{43}\right)$$ $$e\left(\frac{30}{43}\right)$$ $$e\left(\frac{7}{43}\right)$$ $$e\left(\frac{29}{43}\right)$$ $$e\left(\frac{12}{43}\right)$$ $$e\left(\frac{10}{43}\right)$$ $$e\left(\frac{32}{43}\right)$$ $$e\left(\frac{17}{43}\right)$$ $$e\left(\frac{11}{43}\right)$$ $$e\left(\frac{16}{43}\right)$$
$$\chi_{173}(52,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{33}{43}\right)$$ $$e\left(\frac{31}{43}\right)$$ $$e\left(\frac{23}{43}\right)$$ $$e\left(\frac{40}{43}\right)$$ $$e\left(\frac{21}{43}\right)$$ $$e\left(\frac{39}{43}\right)$$ $$e\left(\frac{13}{43}\right)$$ $$e\left(\frac{19}{43}\right)$$ $$e\left(\frac{30}{43}\right)$$ $$e\left(\frac{28}{43}\right)$$
$$\chi_{173}(57,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{15}{43}\right)$$ $$e\left(\frac{18}{43}\right)$$ $$e\left(\frac{30}{43}\right)$$ $$e\left(\frac{26}{43}\right)$$ $$e\left(\frac{33}{43}\right)$$ $$e\left(\frac{6}{43}\right)$$ $$e\left(\frac{2}{43}\right)$$ $$e\left(\frac{36}{43}\right)$$ $$e\left(\frac{41}{43}\right)$$ $$e\left(\frac{1}{43}\right)$$
$$\chi_{173}(60,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{17}{43}\right)$$ $$e\left(\frac{29}{43}\right)$$ $$e\left(\frac{34}{43}\right)$$ $$e\left(\frac{18}{43}\right)$$ $$e\left(\frac{3}{43}\right)$$ $$e\left(\frac{24}{43}\right)$$ $$e\left(\frac{8}{43}\right)$$ $$e\left(\frac{15}{43}\right)$$ $$e\left(\frac{35}{43}\right)$$ $$e\left(\frac{4}{43}\right)$$
$$\chi_{173}(81,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{27}{43}\right)$$ $$e\left(\frac{41}{43}\right)$$ $$e\left(\frac{11}{43}\right)$$ $$e\left(\frac{21}{43}\right)$$ $$e\left(\frac{25}{43}\right)$$ $$e\left(\frac{28}{43}\right)$$ $$e\left(\frac{38}{43}\right)$$ $$e\left(\frac{39}{43}\right)$$ $$e\left(\frac{5}{43}\right)$$ $$e\left(\frac{19}{43}\right)$$
$$\chi_{173}(83,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{2}{43}\right)$$ $$e\left(\frac{11}{43}\right)$$ $$e\left(\frac{4}{43}\right)$$ $$e\left(\frac{35}{43}\right)$$ $$e\left(\frac{13}{43}\right)$$ $$e\left(\frac{18}{43}\right)$$ $$e\left(\frac{6}{43}\right)$$ $$e\left(\frac{22}{43}\right)$$ $$e\left(\frac{37}{43}\right)$$ $$e\left(\frac{3}{43}\right)$$
$$\chi_{173}(84,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{31}{43}\right)$$ $$e\left(\frac{20}{43}\right)$$ $$e\left(\frac{19}{43}\right)$$ $$e\left(\frac{5}{43}\right)$$ $$e\left(\frac{8}{43}\right)$$ $$e\left(\frac{21}{43}\right)$$ $$e\left(\frac{7}{43}\right)$$ $$e\left(\frac{40}{43}\right)$$ $$e\left(\frac{36}{43}\right)$$ $$e\left(\frac{25}{43}\right)$$
$$\chi_{173}(85,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{28}{43}\right)$$ $$e\left(\frac{25}{43}\right)$$ $$e\left(\frac{13}{43}\right)$$ $$e\left(\frac{17}{43}\right)$$ $$e\left(\frac{10}{43}\right)$$ $$e\left(\frac{37}{43}\right)$$ $$e\left(\frac{41}{43}\right)$$ $$e\left(\frac{7}{43}\right)$$ $$e\left(\frac{2}{43}\right)$$ $$e\left(\frac{42}{43}\right)$$
$$\chi_{173}(95,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{18}{43}\right)$$ $$e\left(\frac{13}{43}\right)$$ $$e\left(\frac{36}{43}\right)$$ $$e\left(\frac{14}{43}\right)$$ $$e\left(\frac{31}{43}\right)$$ $$e\left(\frac{33}{43}\right)$$ $$e\left(\frac{11}{43}\right)$$ $$e\left(\frac{26}{43}\right)$$ $$e\left(\frac{32}{43}\right)$$ $$e\left(\frac{27}{43}\right)$$
$$\chi_{173}(96,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{8}{43}\right)$$ $$e\left(\frac{1}{43}\right)$$ $$e\left(\frac{16}{43}\right)$$ $$e\left(\frac{11}{43}\right)$$ $$e\left(\frac{9}{43}\right)$$ $$e\left(\frac{29}{43}\right)$$ $$e\left(\frac{24}{43}\right)$$ $$e\left(\frac{2}{43}\right)$$ $$e\left(\frac{19}{43}\right)$$ $$e\left(\frac{12}{43}\right)$$
$$\chi_{173}(100,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{20}{43}\right)$$ $$e\left(\frac{24}{43}\right)$$ $$e\left(\frac{40}{43}\right)$$ $$e\left(\frac{6}{43}\right)$$ $$e\left(\frac{1}{43}\right)$$ $$e\left(\frac{8}{43}\right)$$ $$e\left(\frac{17}{43}\right)$$ $$e\left(\frac{5}{43}\right)$$ $$e\left(\frac{26}{43}\right)$$ $$e\left(\frac{30}{43}\right)$$
$$\chi_{173}(106,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{39}{43}\right)$$ $$e\left(\frac{21}{43}\right)$$ $$e\left(\frac{35}{43}\right)$$ $$e\left(\frac{16}{43}\right)$$ $$e\left(\frac{17}{43}\right)$$ $$e\left(\frac{7}{43}\right)$$ $$e\left(\frac{31}{43}\right)$$ $$e\left(\frac{42}{43}\right)$$ $$e\left(\frac{12}{43}\right)$$ $$e\left(\frac{37}{43}\right)$$
$$\chi_{173}(109,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{23}{43}\right)$$ $$e\left(\frac{19}{43}\right)$$ $$e\left(\frac{3}{43}\right)$$ $$e\left(\frac{37}{43}\right)$$ $$e\left(\frac{42}{43}\right)$$ $$e\left(\frac{35}{43}\right)$$ $$e\left(\frac{26}{43}\right)$$ $$e\left(\frac{38}{43}\right)$$ $$e\left(\frac{17}{43}\right)$$ $$e\left(\frac{13}{43}\right)$$
$$\chi_{173}(117,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{3}{43}\right)$$ $$e\left(\frac{38}{43}\right)$$ $$e\left(\frac{6}{43}\right)$$ $$e\left(\frac{31}{43}\right)$$ $$e\left(\frac{41}{43}\right)$$ $$e\left(\frac{27}{43}\right)$$ $$e\left(\frac{9}{43}\right)$$ $$e\left(\frac{33}{43}\right)$$ $$e\left(\frac{34}{43}\right)$$ $$e\left(\frac{26}{43}\right)$$
$$\chi_{173}(118,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{37}{43}\right)$$ $$e\left(\frac{10}{43}\right)$$ $$e\left(\frac{31}{43}\right)$$ $$e\left(\frac{24}{43}\right)$$ $$e\left(\frac{4}{43}\right)$$ $$e\left(\frac{32}{43}\right)$$ $$e\left(\frac{25}{43}\right)$$ $$e\left(\frac{20}{43}\right)$$ $$e\left(\frac{18}{43}\right)$$ $$e\left(\frac{34}{43}\right)$$
$$\chi_{173}(119,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{42}{43}\right)$$ $$e\left(\frac{16}{43}\right)$$ $$e\left(\frac{41}{43}\right)$$ $$e\left(\frac{4}{43}\right)$$ $$e\left(\frac{15}{43}\right)$$ $$e\left(\frac{34}{43}\right)$$ $$e\left(\frac{40}{43}\right)$$ $$e\left(\frac{32}{43}\right)$$ $$e\left(\frac{3}{43}\right)$$ $$e\left(\frac{20}{43}\right)$$
$$\chi_{173}(124,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{26}{43}\right)$$ $$e\left(\frac{14}{43}\right)$$ $$e\left(\frac{9}{43}\right)$$ $$e\left(\frac{25}{43}\right)$$ $$e\left(\frac{40}{43}\right)$$ $$e\left(\frac{19}{43}\right)$$ $$e\left(\frac{35}{43}\right)$$ $$e\left(\frac{28}{43}\right)$$ $$e\left(\frac{8}{43}\right)$$ $$e\left(\frac{39}{43}\right)$$
$$\chi_{173}(132,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{13}{43}\right)$$ $$e\left(\frac{7}{43}\right)$$ $$e\left(\frac{26}{43}\right)$$ $$e\left(\frac{34}{43}\right)$$ $$e\left(\frac{20}{43}\right)$$ $$e\left(\frac{31}{43}\right)$$ $$e\left(\frac{39}{43}\right)$$ $$e\left(\frac{14}{43}\right)$$ $$e\left(\frac{4}{43}\right)$$ $$e\left(\frac{41}{43}\right)$$
$$\chi_{173}(133,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{32}{43}\right)$$ $$e\left(\frac{4}{43}\right)$$ $$e\left(\frac{21}{43}\right)$$ $$e\left(\frac{1}{43}\right)$$ $$e\left(\frac{36}{43}\right)$$ $$e\left(\frac{30}{43}\right)$$ $$e\left(\frac{10}{43}\right)$$ $$e\left(\frac{8}{43}\right)$$ $$e\left(\frac{33}{43}\right)$$ $$e\left(\frac{5}{43}\right)$$
$$\chi_{173}(135,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{30}{43}\right)$$ $$e\left(\frac{36}{43}\right)$$ $$e\left(\frac{17}{43}\right)$$ $$e\left(\frac{9}{43}\right)$$ $$e\left(\frac{23}{43}\right)$$ $$e\left(\frac{12}{43}\right)$$ $$e\left(\frac{4}{43}\right)$$ $$e\left(\frac{29}{43}\right)$$ $$e\left(\frac{39}{43}\right)$$ $$e\left(\frac{2}{43}\right)$$
$$\chi_{173}(136,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{19}{43}\right)$$ $$e\left(\frac{40}{43}\right)$$ $$e\left(\frac{38}{43}\right)$$ $$e\left(\frac{10}{43}\right)$$ $$e\left(\frac{16}{43}\right)$$ $$e\left(\frac{42}{43}\right)$$ $$e\left(\frac{14}{43}\right)$$ $$e\left(\frac{37}{43}\right)$$ $$e\left(\frac{29}{43}\right)$$ $$e\left(\frac{7}{43}\right)$$