# Properties

 Label 729.j Modulus $729$ Conductor $243$ Order $162$ Real no Primitive no Minimal no Parity odd

# Related objects

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

H = DirichletGroup(729, base_ring=CyclotomicField(162))

M = H._module

chi = DirichletCharacter(H, M([1]))

chi.galois_orbit()

[g,chi] = znchar(Mod(8,729))

order = charorder(g,chi)

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

## Basic properties

 Modulus: $$729$$ Conductor: $$243$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$162$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from 243.j sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: no Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Related number fields

 Field of values: $\Q(\zeta_{81})$ Fixed field: Number field defined by a degree 162 polynomial (not computed)

## First 31 of 54 characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$11$$ $$13$$ $$14$$ $$16$$
$$\chi_{729}(8,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{162}\right)$$ $$e\left(\frac{1}{81}\right)$$ $$e\left(\frac{23}{162}\right)$$ $$e\left(\frac{35}{81}\right)$$ $$e\left(\frac{1}{54}\right)$$ $$e\left(\frac{4}{27}\right)$$ $$e\left(\frac{121}{162}\right)$$ $$e\left(\frac{4}{81}\right)$$ $$e\left(\frac{71}{162}\right)$$ $$e\left(\frac{2}{81}\right)$$
$$\chi_{729}(17,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{11}{162}\right)$$ $$e\left(\frac{11}{81}\right)$$ $$e\left(\frac{91}{162}\right)$$ $$e\left(\frac{61}{81}\right)$$ $$e\left(\frac{11}{54}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{35}{162}\right)$$ $$e\left(\frac{44}{81}\right)$$ $$e\left(\frac{133}{162}\right)$$ $$e\left(\frac{22}{81}\right)$$
$$\chi_{729}(35,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{139}{162}\right)$$ $$e\left(\frac{58}{81}\right)$$ $$e\left(\frac{119}{162}\right)$$ $$e\left(\frac{5}{81}\right)$$ $$e\left(\frac{31}{54}\right)$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{133}{162}\right)$$ $$e\left(\frac{70}{81}\right)$$ $$e\left(\frac{149}{162}\right)$$ $$e\left(\frac{35}{81}\right)$$
$$\chi_{729}(44,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{95}{162}\right)$$ $$e\left(\frac{14}{81}\right)$$ $$e\left(\frac{79}{162}\right)$$ $$e\left(\frac{4}{81}\right)$$ $$e\left(\frac{41}{54}\right)$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{155}{162}\right)$$ $$e\left(\frac{56}{81}\right)$$ $$e\left(\frac{103}{162}\right)$$ $$e\left(\frac{28}{81}\right)$$
$$\chi_{729}(62,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{115}{162}\right)$$ $$e\left(\frac{34}{81}\right)$$ $$e\left(\frac{53}{162}\right)$$ $$e\left(\frac{56}{81}\right)$$ $$e\left(\frac{7}{54}\right)$$ $$e\left(\frac{1}{27}\right)$$ $$e\left(\frac{145}{162}\right)$$ $$e\left(\frac{55}{81}\right)$$ $$e\left(\frac{65}{162}\right)$$ $$e\left(\frac{68}{81}\right)$$
$$\chi_{729}(71,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{17}{162}\right)$$ $$e\left(\frac{17}{81}\right)$$ $$e\left(\frac{67}{162}\right)$$ $$e\left(\frac{28}{81}\right)$$ $$e\left(\frac{17}{54}\right)$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{113}{162}\right)$$ $$e\left(\frac{68}{81}\right)$$ $$e\left(\frac{73}{162}\right)$$ $$e\left(\frac{34}{81}\right)$$
$$\chi_{729}(89,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{91}{162}\right)$$ $$e\left(\frac{10}{81}\right)$$ $$e\left(\frac{149}{162}\right)$$ $$e\left(\frac{26}{81}\right)$$ $$e\left(\frac{37}{54}\right)$$ $$e\left(\frac{13}{27}\right)$$ $$e\left(\frac{157}{162}\right)$$ $$e\left(\frac{40}{81}\right)$$ $$e\left(\frac{143}{162}\right)$$ $$e\left(\frac{20}{81}\right)$$
$$\chi_{729}(98,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{101}{162}\right)$$ $$e\left(\frac{20}{81}\right)$$ $$e\left(\frac{55}{162}\right)$$ $$e\left(\frac{52}{81}\right)$$ $$e\left(\frac{47}{54}\right)$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{71}{162}\right)$$ $$e\left(\frac{80}{81}\right)$$ $$e\left(\frac{43}{162}\right)$$ $$e\left(\frac{40}{81}\right)$$
$$\chi_{729}(116,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{67}{162}\right)$$ $$e\left(\frac{67}{81}\right)$$ $$e\left(\frac{83}{162}\right)$$ $$e\left(\frac{77}{81}\right)$$ $$e\left(\frac{13}{54}\right)$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{7}{162}\right)$$ $$e\left(\frac{25}{81}\right)$$ $$e\left(\frac{59}{162}\right)$$ $$e\left(\frac{53}{81}\right)$$
$$\chi_{729}(125,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{23}{162}\right)$$ $$e\left(\frac{23}{81}\right)$$ $$e\left(\frac{43}{162}\right)$$ $$e\left(\frac{76}{81}\right)$$ $$e\left(\frac{23}{54}\right)$$ $$e\left(\frac{11}{27}\right)$$ $$e\left(\frac{29}{162}\right)$$ $$e\left(\frac{11}{81}\right)$$ $$e\left(\frac{13}{162}\right)$$ $$e\left(\frac{46}{81}\right)$$
$$\chi_{729}(143,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{43}{162}\right)$$ $$e\left(\frac{43}{81}\right)$$ $$e\left(\frac{17}{162}\right)$$ $$e\left(\frac{47}{81}\right)$$ $$e\left(\frac{43}{54}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{19}{162}\right)$$ $$e\left(\frac{10}{81}\right)$$ $$e\left(\frac{137}{162}\right)$$ $$e\left(\frac{5}{81}\right)$$
$$\chi_{729}(152,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{107}{162}\right)$$ $$e\left(\frac{26}{81}\right)$$ $$e\left(\frac{31}{162}\right)$$ $$e\left(\frac{19}{81}\right)$$ $$e\left(\frac{53}{54}\right)$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{149}{162}\right)$$ $$e\left(\frac{23}{81}\right)$$ $$e\left(\frac{145}{162}\right)$$ $$e\left(\frac{52}{81}\right)$$
$$\chi_{729}(170,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{19}{162}\right)$$ $$e\left(\frac{19}{81}\right)$$ $$e\left(\frac{113}{162}\right)$$ $$e\left(\frac{17}{81}\right)$$ $$e\left(\frac{19}{54}\right)$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{31}{162}\right)$$ $$e\left(\frac{76}{81}\right)$$ $$e\left(\frac{53}{162}\right)$$ $$e\left(\frac{38}{81}\right)$$
$$\chi_{729}(179,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{29}{162}\right)$$ $$e\left(\frac{29}{81}\right)$$ $$e\left(\frac{19}{162}\right)$$ $$e\left(\frac{43}{81}\right)$$ $$e\left(\frac{29}{54}\right)$$ $$e\left(\frac{8}{27}\right)$$ $$e\left(\frac{107}{162}\right)$$ $$e\left(\frac{35}{81}\right)$$ $$e\left(\frac{115}{162}\right)$$ $$e\left(\frac{58}{81}\right)$$
$$\chi_{729}(197,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{157}{162}\right)$$ $$e\left(\frac{76}{81}\right)$$ $$e\left(\frac{47}{162}\right)$$ $$e\left(\frac{68}{81}\right)$$ $$e\left(\frac{49}{54}\right)$$ $$e\left(\frac{7}{27}\right)$$ $$e\left(\frac{43}{162}\right)$$ $$e\left(\frac{61}{81}\right)$$ $$e\left(\frac{131}{162}\right)$$ $$e\left(\frac{71}{81}\right)$$
$$\chi_{729}(206,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{113}{162}\right)$$ $$e\left(\frac{32}{81}\right)$$ $$e\left(\frac{7}{162}\right)$$ $$e\left(\frac{67}{81}\right)$$ $$e\left(\frac{5}{54}\right)$$ $$e\left(\frac{20}{27}\right)$$ $$e\left(\frac{65}{162}\right)$$ $$e\left(\frac{47}{81}\right)$$ $$e\left(\frac{85}{162}\right)$$ $$e\left(\frac{64}{81}\right)$$
$$\chi_{729}(224,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{133}{162}\right)$$ $$e\left(\frac{52}{81}\right)$$ $$e\left(\frac{143}{162}\right)$$ $$e\left(\frac{38}{81}\right)$$ $$e\left(\frac{25}{54}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{55}{162}\right)$$ $$e\left(\frac{46}{81}\right)$$ $$e\left(\frac{47}{162}\right)$$ $$e\left(\frac{23}{81}\right)$$
$$\chi_{729}(233,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{35}{162}\right)$$ $$e\left(\frac{35}{81}\right)$$ $$e\left(\frac{157}{162}\right)$$ $$e\left(\frac{10}{81}\right)$$ $$e\left(\frac{35}{54}\right)$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{23}{162}\right)$$ $$e\left(\frac{59}{81}\right)$$ $$e\left(\frac{55}{162}\right)$$ $$e\left(\frac{70}{81}\right)$$
$$\chi_{729}(251,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{109}{162}\right)$$ $$e\left(\frac{28}{81}\right)$$ $$e\left(\frac{77}{162}\right)$$ $$e\left(\frac{8}{81}\right)$$ $$e\left(\frac{1}{54}\right)$$ $$e\left(\frac{4}{27}\right)$$ $$e\left(\frac{67}{162}\right)$$ $$e\left(\frac{31}{81}\right)$$ $$e\left(\frac{125}{162}\right)$$ $$e\left(\frac{56}{81}\right)$$
$$\chi_{729}(260,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{119}{162}\right)$$ $$e\left(\frac{38}{81}\right)$$ $$e\left(\frac{145}{162}\right)$$ $$e\left(\frac{34}{81}\right)$$ $$e\left(\frac{11}{54}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{143}{162}\right)$$ $$e\left(\frac{71}{81}\right)$$ $$e\left(\frac{25}{162}\right)$$ $$e\left(\frac{76}{81}\right)$$
$$\chi_{729}(278,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{85}{162}\right)$$ $$e\left(\frac{4}{81}\right)$$ $$e\left(\frac{11}{162}\right)$$ $$e\left(\frac{59}{81}\right)$$ $$e\left(\frac{31}{54}\right)$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{79}{162}\right)$$ $$e\left(\frac{16}{81}\right)$$ $$e\left(\frac{41}{162}\right)$$ $$e\left(\frac{8}{81}\right)$$
$$\chi_{729}(287,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{41}{162}\right)$$ $$e\left(\frac{41}{81}\right)$$ $$e\left(\frac{133}{162}\right)$$ $$e\left(\frac{58}{81}\right)$$ $$e\left(\frac{41}{54}\right)$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{101}{162}\right)$$ $$e\left(\frac{2}{81}\right)$$ $$e\left(\frac{157}{162}\right)$$ $$e\left(\frac{1}{81}\right)$$
$$\chi_{729}(305,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{61}{162}\right)$$ $$e\left(\frac{61}{81}\right)$$ $$e\left(\frac{107}{162}\right)$$ $$e\left(\frac{29}{81}\right)$$ $$e\left(\frac{7}{54}\right)$$ $$e\left(\frac{1}{27}\right)$$ $$e\left(\frac{91}{162}\right)$$ $$e\left(\frac{1}{81}\right)$$ $$e\left(\frac{119}{162}\right)$$ $$e\left(\frac{41}{81}\right)$$
$$\chi_{729}(314,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{125}{162}\right)$$ $$e\left(\frac{44}{81}\right)$$ $$e\left(\frac{121}{162}\right)$$ $$e\left(\frac{1}{81}\right)$$ $$e\left(\frac{17}{54}\right)$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{59}{162}\right)$$ $$e\left(\frac{14}{81}\right)$$ $$e\left(\frac{127}{162}\right)$$ $$e\left(\frac{7}{81}\right)$$
$$\chi_{729}(332,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{37}{162}\right)$$ $$e\left(\frac{37}{81}\right)$$ $$e\left(\frac{41}{162}\right)$$ $$e\left(\frac{80}{81}\right)$$ $$e\left(\frac{37}{54}\right)$$ $$e\left(\frac{13}{27}\right)$$ $$e\left(\frac{103}{162}\right)$$ $$e\left(\frac{67}{81}\right)$$ $$e\left(\frac{35}{162}\right)$$ $$e\left(\frac{74}{81}\right)$$
$$\chi_{729}(341,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{47}{162}\right)$$ $$e\left(\frac{47}{81}\right)$$ $$e\left(\frac{109}{162}\right)$$ $$e\left(\frac{25}{81}\right)$$ $$e\left(\frac{47}{54}\right)$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{17}{162}\right)$$ $$e\left(\frac{26}{81}\right)$$ $$e\left(\frac{97}{162}\right)$$ $$e\left(\frac{13}{81}\right)$$
$$\chi_{729}(359,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{13}{162}\right)$$ $$e\left(\frac{13}{81}\right)$$ $$e\left(\frac{137}{162}\right)$$ $$e\left(\frac{50}{81}\right)$$ $$e\left(\frac{13}{54}\right)$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{115}{162}\right)$$ $$e\left(\frac{52}{81}\right)$$ $$e\left(\frac{113}{162}\right)$$ $$e\left(\frac{26}{81}\right)$$
$$\chi_{729}(368,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{131}{162}\right)$$ $$e\left(\frac{50}{81}\right)$$ $$e\left(\frac{97}{162}\right)$$ $$e\left(\frac{49}{81}\right)$$ $$e\left(\frac{23}{54}\right)$$ $$e\left(\frac{11}{27}\right)$$ $$e\left(\frac{137}{162}\right)$$ $$e\left(\frac{38}{81}\right)$$ $$e\left(\frac{67}{162}\right)$$ $$e\left(\frac{19}{81}\right)$$
$$\chi_{729}(386,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{151}{162}\right)$$ $$e\left(\frac{70}{81}\right)$$ $$e\left(\frac{71}{162}\right)$$ $$e\left(\frac{20}{81}\right)$$ $$e\left(\frac{43}{54}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{127}{162}\right)$$ $$e\left(\frac{37}{81}\right)$$ $$e\left(\frac{29}{162}\right)$$ $$e\left(\frac{59}{81}\right)$$
$$\chi_{729}(395,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{53}{162}\right)$$ $$e\left(\frac{53}{81}\right)$$ $$e\left(\frac{85}{162}\right)$$ $$e\left(\frac{73}{81}\right)$$ $$e\left(\frac{53}{54}\right)$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{95}{162}\right)$$ $$e\left(\frac{50}{81}\right)$$ $$e\left(\frac{37}{162}\right)$$ $$e\left(\frac{25}{81}\right)$$
$$\chi_{729}(413,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{127}{162}\right)$$ $$e\left(\frac{46}{81}\right)$$ $$e\left(\frac{5}{162}\right)$$ $$e\left(\frac{71}{81}\right)$$ $$e\left(\frac{19}{54}\right)$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{139}{162}\right)$$ $$e\left(\frac{22}{81}\right)$$ $$e\left(\frac{107}{162}\right)$$ $$e\left(\frac{11}{81}\right)$$