# Properties

 Label 729.l Modulus $729$ Conductor $729$ Order $486$ Real no Primitive yes Minimal yes Parity odd

# Related objects

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

H = DirichletGroup(729, base_ring=CyclotomicField(486))

M = H._module

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

chi.galois_orbit()

[g,chi] = znchar(Mod(2,729))

order = charorder(g,chi)

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

## Basic properties

 Modulus: $$729$$ Conductor: $$729$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$486$$ 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: $\Q(\zeta_{243})$ Fixed field: Number field defined by a degree 486 polynomial (not computed)

## First 31 of 162 characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$11$$ $$13$$ $$14$$ $$16$$
$$\chi_{729}(2,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{486}\right)$$ $$e\left(\frac{1}{243}\right)$$ $$e\left(\frac{23}{486}\right)$$ $$e\left(\frac{197}{243}\right)$$ $$e\left(\frac{1}{162}\right)$$ $$e\left(\frac{4}{81}\right)$$ $$e\left(\frac{283}{486}\right)$$ $$e\left(\frac{166}{243}\right)$$ $$e\left(\frac{395}{486}\right)$$ $$e\left(\frac{2}{243}\right)$$
$$\chi_{729}(5,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{23}{486}\right)$$ $$e\left(\frac{23}{243}\right)$$ $$e\left(\frac{43}{486}\right)$$ $$e\left(\frac{157}{243}\right)$$ $$e\left(\frac{23}{162}\right)$$ $$e\left(\frac{11}{81}\right)$$ $$e\left(\frac{191}{486}\right)$$ $$e\left(\frac{173}{243}\right)$$ $$e\left(\frac{337}{486}\right)$$ $$e\left(\frac{46}{243}\right)$$
$$\chi_{729}(11,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{283}{486}\right)$$ $$e\left(\frac{40}{243}\right)$$ $$e\left(\frac{191}{486}\right)$$ $$e\left(\frac{104}{243}\right)$$ $$e\left(\frac{121}{162}\right)$$ $$e\left(\frac{79}{81}\right)$$ $$e\left(\frac{385}{486}\right)$$ $$e\left(\frac{79}{243}\right)$$ $$e\left(\frac{5}{486}\right)$$ $$e\left(\frac{80}{243}\right)$$
$$\chi_{729}(14,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{395}{486}\right)$$ $$e\left(\frac{152}{243}\right)$$ $$e\left(\frac{337}{486}\right)$$ $$e\left(\frac{55}{243}\right)$$ $$e\left(\frac{71}{162}\right)$$ $$e\left(\frac{41}{81}\right)$$ $$e\left(\frac{5}{486}\right)$$ $$e\left(\frac{203}{243}\right)$$ $$e\left(\frac{19}{486}\right)$$ $$e\left(\frac{61}{243}\right)$$
$$\chi_{729}(20,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{25}{486}\right)$$ $$e\left(\frac{25}{243}\right)$$ $$e\left(\frac{89}{486}\right)$$ $$e\left(\frac{65}{243}\right)$$ $$e\left(\frac{25}{162}\right)$$ $$e\left(\frac{19}{81}\right)$$ $$e\left(\frac{271}{486}\right)$$ $$e\left(\frac{19}{243}\right)$$ $$e\left(\frac{155}{486}\right)$$ $$e\left(\frac{50}{243}\right)$$
$$\chi_{729}(23,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{389}{486}\right)$$ $$e\left(\frac{146}{243}\right)$$ $$e\left(\frac{199}{486}\right)$$ $$e\left(\frac{88}{243}\right)$$ $$e\left(\frac{65}{162}\right)$$ $$e\left(\frac{17}{81}\right)$$ $$e\left(\frac{251}{486}\right)$$ $$e\left(\frac{179}{243}\right)$$ $$e\left(\frac{79}{486}\right)$$ $$e\left(\frac{49}{243}\right)$$
$$\chi_{729}(29,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{199}{486}\right)$$ $$e\left(\frac{199}{243}\right)$$ $$e\left(\frac{203}{486}\right)$$ $$e\left(\frac{80}{243}\right)$$ $$e\left(\frac{37}{162}\right)$$ $$e\left(\frac{67}{81}\right)$$ $$e\left(\frac{427}{486}\right)$$ $$e\left(\frac{229}{243}\right)$$ $$e\left(\frac{359}{486}\right)$$ $$e\left(\frac{155}{243}\right)$$
$$\chi_{729}(32,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{5}{486}\right)$$ $$e\left(\frac{5}{243}\right)$$ $$e\left(\frac{115}{486}\right)$$ $$e\left(\frac{13}{243}\right)$$ $$e\left(\frac{5}{162}\right)$$ $$e\left(\frac{20}{81}\right)$$ $$e\left(\frac{443}{486}\right)$$ $$e\left(\frac{101}{243}\right)$$ $$e\left(\frac{31}{486}\right)$$ $$e\left(\frac{10}{243}\right)$$
$$\chi_{729}(38,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{319}{486}\right)$$ $$e\left(\frac{76}{243}\right)$$ $$e\left(\frac{47}{486}\right)$$ $$e\left(\frac{149}{243}\right)$$ $$e\left(\frac{157}{162}\right)$$ $$e\left(\frac{61}{81}\right)$$ $$e\left(\frac{367}{486}\right)$$ $$e\left(\frac{223}{243}\right)$$ $$e\left(\frac{131}{486}\right)$$ $$e\left(\frac{152}{243}\right)$$
$$\chi_{729}(41,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{215}{486}\right)$$ $$e\left(\frac{215}{243}\right)$$ $$e\left(\frac{85}{486}\right)$$ $$e\left(\frac{73}{243}\right)$$ $$e\left(\frac{53}{162}\right)$$ $$e\left(\frac{50}{81}\right)$$ $$e\left(\frac{95}{486}\right)$$ $$e\left(\frac{212}{243}\right)$$ $$e\left(\frac{361}{486}\right)$$ $$e\left(\frac{187}{243}\right)$$
$$\chi_{729}(47,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{385}{486}\right)$$ $$e\left(\frac{142}{243}\right)$$ $$e\left(\frac{107}{486}\right)$$ $$e\left(\frac{29}{243}\right)$$ $$e\left(\frac{61}{162}\right)$$ $$e\left(\frac{1}{81}\right)$$ $$e\left(\frac{91}{486}\right)$$ $$e\left(\frac{1}{243}\right)$$ $$e\left(\frac{443}{486}\right)$$ $$e\left(\frac{41}{243}\right)$$
$$\chi_{729}(50,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{47}{486}\right)$$ $$e\left(\frac{47}{243}\right)$$ $$e\left(\frac{109}{486}\right)$$ $$e\left(\frac{25}{243}\right)$$ $$e\left(\frac{47}{162}\right)$$ $$e\left(\frac{26}{81}\right)$$ $$e\left(\frac{179}{486}\right)$$ $$e\left(\frac{26}{243}\right)$$ $$e\left(\frac{97}{486}\right)$$ $$e\left(\frac{94}{243}\right)$$
$$\chi_{729}(56,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{397}{486}\right)$$ $$e\left(\frac{154}{243}\right)$$ $$e\left(\frac{383}{486}\right)$$ $$e\left(\frac{206}{243}\right)$$ $$e\left(\frac{73}{162}\right)$$ $$e\left(\frac{49}{81}\right)$$ $$e\left(\frac{85}{486}\right)$$ $$e\left(\frac{49}{243}\right)$$ $$e\left(\frac{323}{486}\right)$$ $$e\left(\frac{65}{243}\right)$$
$$\chi_{729}(59,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{473}{486}\right)$$ $$e\left(\frac{230}{243}\right)$$ $$e\left(\frac{187}{486}\right)$$ $$e\left(\frac{112}{243}\right)$$ $$e\left(\frac{149}{162}\right)$$ $$e\left(\frac{29}{81}\right)$$ $$e\left(\frac{209}{486}\right)$$ $$e\left(\frac{29}{243}\right)$$ $$e\left(\frac{211}{486}\right)$$ $$e\left(\frac{217}{243}\right)$$
$$\chi_{729}(65,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{355}{486}\right)$$ $$e\left(\frac{112}{243}\right)$$ $$e\left(\frac{389}{486}\right)$$ $$e\left(\frac{194}{243}\right)$$ $$e\left(\frac{31}{162}\right)$$ $$e\left(\frac{43}{81}\right)$$ $$e\left(\frac{349}{486}\right)$$ $$e\left(\frac{124}{243}\right)$$ $$e\left(\frac{257}{486}\right)$$ $$e\left(\frac{224}{243}\right)$$
$$\chi_{729}(68,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{35}{486}\right)$$ $$e\left(\frac{35}{243}\right)$$ $$e\left(\frac{319}{486}\right)$$ $$e\left(\frac{91}{243}\right)$$ $$e\left(\frac{35}{162}\right)$$ $$e\left(\frac{59}{81}\right)$$ $$e\left(\frac{185}{486}\right)$$ $$e\left(\frac{221}{243}\right)$$ $$e\left(\frac{217}{486}\right)$$ $$e\left(\frac{70}{243}\right)$$
$$\chi_{729}(74,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{259}{486}\right)$$ $$e\left(\frac{16}{243}\right)$$ $$e\left(\frac{125}{486}\right)$$ $$e\left(\frac{236}{243}\right)$$ $$e\left(\frac{97}{162}\right)$$ $$e\left(\frac{64}{81}\right)$$ $$e\left(\frac{397}{486}\right)$$ $$e\left(\frac{226}{243}\right)$$ $$e\left(\frac{245}{486}\right)$$ $$e\left(\frac{32}{243}\right)$$
$$\chi_{729}(77,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{191}{486}\right)$$ $$e\left(\frac{191}{243}\right)$$ $$e\left(\frac{19}{486}\right)$$ $$e\left(\frac{205}{243}\right)$$ $$e\left(\frac{29}{162}\right)$$ $$e\left(\frac{35}{81}\right)$$ $$e\left(\frac{107}{486}\right)$$ $$e\left(\frac{116}{243}\right)$$ $$e\left(\frac{115}{486}\right)$$ $$e\left(\frac{139}{243}\right)$$
$$\chi_{729}(83,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{109}{486}\right)$$ $$e\left(\frac{109}{243}\right)$$ $$e\left(\frac{77}{486}\right)$$ $$e\left(\frac{89}{243}\right)$$ $$e\left(\frac{109}{162}\right)$$ $$e\left(\frac{31}{81}\right)$$ $$e\left(\frac{229}{486}\right)$$ $$e\left(\frac{112}{243}\right)$$ $$e\left(\frac{287}{486}\right)$$ $$e\left(\frac{218}{243}\right)$$
$$\chi_{729}(86,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{455}{486}\right)$$ $$e\left(\frac{212}{243}\right)$$ $$e\left(\frac{259}{486}\right)$$ $$e\left(\frac{211}{243}\right)$$ $$e\left(\frac{131}{162}\right)$$ $$e\left(\frac{38}{81}\right)$$ $$e\left(\frac{461}{486}\right)$$ $$e\left(\frac{200}{243}\right)$$ $$e\left(\frac{391}{486}\right)$$ $$e\left(\frac{181}{243}\right)$$
$$\chi_{729}(92,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{391}{486}\right)$$ $$e\left(\frac{148}{243}\right)$$ $$e\left(\frac{245}{486}\right)$$ $$e\left(\frac{239}{243}\right)$$ $$e\left(\frac{67}{162}\right)$$ $$e\left(\frac{25}{81}\right)$$ $$e\left(\frac{331}{486}\right)$$ $$e\left(\frac{25}{243}\right)$$ $$e\left(\frac{383}{486}\right)$$ $$e\left(\frac{53}{243}\right)$$
$$\chi_{729}(95,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{341}{486}\right)$$ $$e\left(\frac{98}{243}\right)$$ $$e\left(\frac{67}{486}\right)$$ $$e\left(\frac{109}{243}\right)$$ $$e\left(\frac{17}{162}\right)$$ $$e\left(\frac{68}{81}\right)$$ $$e\left(\frac{275}{486}\right)$$ $$e\left(\frac{230}{243}\right)$$ $$e\left(\frac{73}{486}\right)$$ $$e\left(\frac{196}{243}\right)$$
$$\chi_{729}(101,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{133}{486}\right)$$ $$e\left(\frac{133}{243}\right)$$ $$e\left(\frac{143}{486}\right)$$ $$e\left(\frac{200}{243}\right)$$ $$e\left(\frac{133}{162}\right)$$ $$e\left(\frac{46}{81}\right)$$ $$e\left(\frac{217}{486}\right)$$ $$e\left(\frac{208}{243}\right)$$ $$e\left(\frac{47}{486}\right)$$ $$e\left(\frac{23}{243}\right)$$
$$\chi_{729}(104,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{335}{486}\right)$$ $$e\left(\frac{92}{243}\right)$$ $$e\left(\frac{415}{486}\right)$$ $$e\left(\frac{142}{243}\right)$$ $$e\left(\frac{11}{162}\right)$$ $$e\left(\frac{44}{81}\right)$$ $$e\left(\frac{35}{486}\right)$$ $$e\left(\frac{206}{243}\right)$$ $$e\left(\frac{133}{486}\right)$$ $$e\left(\frac{184}{243}\right)$$
$$\chi_{729}(110,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{307}{486}\right)$$ $$e\left(\frac{64}{243}\right)$$ $$e\left(\frac{257}{486}\right)$$ $$e\left(\frac{215}{243}\right)$$ $$e\left(\frac{145}{162}\right)$$ $$e\left(\frac{13}{81}\right)$$ $$e\left(\frac{373}{486}\right)$$ $$e\left(\frac{175}{243}\right)$$ $$e\left(\frac{251}{486}\right)$$ $$e\left(\frac{128}{243}\right)$$
$$\chi_{729}(113,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{437}{486}\right)$$ $$e\left(\frac{194}{243}\right)$$ $$e\left(\frac{331}{486}\right)$$ $$e\left(\frac{67}{243}\right)$$ $$e\left(\frac{113}{162}\right)$$ $$e\left(\frac{47}{81}\right)$$ $$e\left(\frac{227}{486}\right)$$ $$e\left(\frac{128}{243}\right)$$ $$e\left(\frac{85}{486}\right)$$ $$e\left(\frac{145}{243}\right)$$
$$\chi_{729}(119,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{427}{486}\right)$$ $$e\left(\frac{184}{243}\right)$$ $$e\left(\frac{101}{486}\right)$$ $$e\left(\frac{41}{243}\right)$$ $$e\left(\frac{103}{162}\right)$$ $$e\left(\frac{7}{81}\right)$$ $$e\left(\frac{313}{486}\right)$$ $$e\left(\frac{169}{243}\right)$$ $$e\left(\frac{23}{486}\right)$$ $$e\left(\frac{125}{243}\right)$$
$$\chi_{729}(122,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{161}{486}\right)$$ $$e\left(\frac{161}{243}\right)$$ $$e\left(\frac{301}{486}\right)$$ $$e\left(\frac{127}{243}\right)$$ $$e\left(\frac{161}{162}\right)$$ $$e\left(\frac{77}{81}\right)$$ $$e\left(\frac{365}{486}\right)$$ $$e\left(\frac{239}{243}\right)$$ $$e\left(\frac{415}{486}\right)$$ $$e\left(\frac{79}{243}\right)$$
$$\chi_{729}(128,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{7}{486}\right)$$ $$e\left(\frac{7}{243}\right)$$ $$e\left(\frac{161}{486}\right)$$ $$e\left(\frac{164}{243}\right)$$ $$e\left(\frac{7}{162}\right)$$ $$e\left(\frac{28}{81}\right)$$ $$e\left(\frac{37}{486}\right)$$ $$e\left(\frac{190}{243}\right)$$ $$e\left(\frac{335}{486}\right)$$ $$e\left(\frac{14}{243}\right)$$
$$\chi_{729}(131,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{479}{486}\right)$$ $$e\left(\frac{236}{243}\right)$$ $$e\left(\frac{325}{486}\right)$$ $$e\left(\frac{79}{243}\right)$$ $$e\left(\frac{155}{162}\right)$$ $$e\left(\frac{53}{81}\right)$$ $$e\left(\frac{449}{486}\right)$$ $$e\left(\frac{53}{243}\right)$$ $$e\left(\frac{151}{486}\right)$$ $$e\left(\frac{229}{243}\right)$$
$$\chi_{729}(137,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{19}{486}\right)$$ $$e\left(\frac{19}{243}\right)$$ $$e\left(\frac{437}{486}\right)$$ $$e\left(\frac{98}{243}\right)$$ $$e\left(\frac{19}{162}\right)$$ $$e\left(\frac{76}{81}\right)$$ $$e\left(\frac{31}{486}\right)$$ $$e\left(\frac{238}{243}\right)$$ $$e\left(\frac{215}{486}\right)$$ $$e\left(\frac{38}{243}\right)$$