# Properties

 Label 3381.ch Modulus $3381$ Conductor $1127$ Order $462$ Real no Primitive no Minimal yes Parity odd

# Related objects

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

sage: H = DirichletGroup(3381, base_ring=CyclotomicField(462))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,352,441]))

sage: chi.galois_orbit()

pari: [g,chi] = znchar(Mod(37,3381))

pari: order = charorder(g,chi)

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

## Basic properties

 Modulus: $$3381$$ Conductor: $$1127$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$462$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from 1127.bd 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: $\Q(\zeta_{231})$ Fixed field: Number field defined by a degree 462 polynomial (not computed)

## First 31 of 120 characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$8$$ $$10$$ $$11$$ $$13$$ $$16$$ $$17$$ $$19$$
$$\chi_{3381}(37,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{166}{231}\right)$$ $$e\left(\frac{101}{231}\right)$$ $$e\left(\frac{23}{462}\right)$$ $$e\left(\frac{12}{77}\right)$$ $$e\left(\frac{355}{462}\right)$$ $$e\left(\frac{31}{462}\right)$$ $$e\left(\frac{39}{77}\right)$$ $$e\left(\frac{202}{231}\right)$$ $$e\left(\frac{337}{462}\right)$$ $$e\left(\frac{65}{66}\right)$$
$$\chi_{3381}(88,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{95}{231}\right)$$ $$e\left(\frac{190}{231}\right)$$ $$e\left(\frac{73}{462}\right)$$ $$e\left(\frac{18}{77}\right)$$ $$e\left(\frac{263}{462}\right)$$ $$e\left(\frac{239}{462}\right)$$ $$e\left(\frac{20}{77}\right)$$ $$e\left(\frac{149}{231}\right)$$ $$e\left(\frac{5}{462}\right)$$ $$e\left(\frac{37}{66}\right)$$
$$\chi_{3381}(109,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{92}{231}\right)$$ $$e\left(\frac{184}{231}\right)$$ $$e\left(\frac{433}{462}\right)$$ $$e\left(\frac{15}{77}\right)$$ $$e\left(\frac{155}{462}\right)$$ $$e\left(\frac{443}{462}\right)$$ $$e\left(\frac{68}{77}\right)$$ $$e\left(\frac{137}{231}\right)$$ $$e\left(\frac{17}{462}\right)$$ $$e\left(\frac{7}{66}\right)$$
$$\chi_{3381}(130,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{5}{231}\right)$$ $$e\left(\frac{10}{231}\right)$$ $$e\left(\frac{247}{462}\right)$$ $$e\left(\frac{5}{77}\right)$$ $$e\left(\frac{257}{462}\right)$$ $$e\left(\frac{353}{462}\right)$$ $$e\left(\frac{74}{77}\right)$$ $$e\left(\frac{20}{231}\right)$$ $$e\left(\frac{365}{462}\right)$$ $$e\left(\frac{61}{66}\right)$$
$$\chi_{3381}(172,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{167}{231}\right)$$ $$e\left(\frac{103}{231}\right)$$ $$e\left(\frac{211}{462}\right)$$ $$e\left(\frac{13}{77}\right)$$ $$e\left(\frac{83}{462}\right)$$ $$e\left(\frac{425}{462}\right)$$ $$e\left(\frac{23}{77}\right)$$ $$e\left(\frac{206}{231}\right)$$ $$e\left(\frac{179}{462}\right)$$ $$e\left(\frac{31}{66}\right)$$
$$\chi_{3381}(205,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{97}{231}\right)$$ $$e\left(\frac{194}{231}\right)$$ $$e\left(\frac{449}{462}\right)$$ $$e\left(\frac{20}{77}\right)$$ $$e\left(\frac{181}{462}\right)$$ $$e\left(\frac{103}{462}\right)$$ $$e\left(\frac{65}{77}\right)$$ $$e\left(\frac{157}{231}\right)$$ $$e\left(\frac{151}{462}\right)$$ $$e\left(\frac{35}{66}\right)$$
$$\chi_{3381}(235,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{32}{231}\right)$$ $$e\left(\frac{64}{231}\right)$$ $$e\left(\frac{241}{462}\right)$$ $$e\left(\frac{32}{77}\right)$$ $$e\left(\frac{305}{462}\right)$$ $$e\left(\frac{365}{462}\right)$$ $$e\left(\frac{27}{77}\right)$$ $$e\left(\frac{128}{231}\right)$$ $$e\left(\frac{257}{462}\right)$$ $$e\left(\frac{1}{66}\right)$$
$$\chi_{3381}(247,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{169}{231}\right)$$ $$e\left(\frac{107}{231}\right)$$ $$e\left(\frac{125}{462}\right)$$ $$e\left(\frac{15}{77}\right)$$ $$e\left(\frac{1}{462}\right)$$ $$e\left(\frac{289}{462}\right)$$ $$e\left(\frac{68}{77}\right)$$ $$e\left(\frac{214}{231}\right)$$ $$e\left(\frac{325}{462}\right)$$ $$e\left(\frac{29}{66}\right)$$
$$\chi_{3381}(268,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{16}{231}\right)$$ $$e\left(\frac{32}{231}\right)$$ $$e\left(\frac{5}{462}\right)$$ $$e\left(\frac{16}{77}\right)$$ $$e\left(\frac{37}{462}\right)$$ $$e\left(\frac{67}{462}\right)$$ $$e\left(\frac{52}{77}\right)$$ $$e\left(\frac{64}{231}\right)$$ $$e\left(\frac{13}{462}\right)$$ $$e\left(\frac{17}{66}\right)$$
$$\chi_{3381}(310,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{46}{231}\right)$$ $$e\left(\frac{92}{231}\right)$$ $$e\left(\frac{101}{462}\right)$$ $$e\left(\frac{46}{77}\right)$$ $$e\left(\frac{193}{462}\right)$$ $$e\left(\frac{337}{462}\right)$$ $$e\left(\frac{34}{77}\right)$$ $$e\left(\frac{184}{231}\right)$$ $$e\left(\frac{355}{462}\right)$$ $$e\left(\frac{53}{66}\right)$$
$$\chi_{3381}(319,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{83}{231}\right)$$ $$e\left(\frac{166}{231}\right)$$ $$e\left(\frac{127}{462}\right)$$ $$e\left(\frac{6}{77}\right)$$ $$e\left(\frac{293}{462}\right)$$ $$e\left(\frac{131}{462}\right)$$ $$e\left(\frac{58}{77}\right)$$ $$e\left(\frac{101}{231}\right)$$ $$e\left(\frac{53}{462}\right)$$ $$e\left(\frac{49}{66}\right)$$
$$\chi_{3381}(352,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{223}{231}\right)$$ $$e\left(\frac{215}{231}\right)$$ $$e\left(\frac{113}{462}\right)$$ $$e\left(\frac{69}{77}\right)$$ $$e\left(\frac{97}{462}\right)$$ $$e\left(\frac{313}{462}\right)$$ $$e\left(\frac{51}{77}\right)$$ $$e\left(\frac{199}{231}\right)$$ $$e\left(\frac{109}{462}\right)$$ $$e\left(\frac{41}{66}\right)$$
$$\chi_{3381}(382,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{221}{231}\right)$$ $$e\left(\frac{211}{231}\right)$$ $$e\left(\frac{199}{462}\right)$$ $$e\left(\frac{67}{77}\right)$$ $$e\left(\frac{179}{462}\right)$$ $$e\left(\frac{449}{462}\right)$$ $$e\left(\frac{6}{77}\right)$$ $$e\left(\frac{191}{231}\right)$$ $$e\left(\frac{425}{462}\right)$$ $$e\left(\frac{43}{66}\right)$$
$$\chi_{3381}(424,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{173}{231}\right)$$ $$e\left(\frac{115}{231}\right)$$ $$e\left(\frac{415}{462}\right)$$ $$e\left(\frac{19}{77}\right)$$ $$e\left(\frac{299}{462}\right)$$ $$e\left(\frac{17}{462}\right)$$ $$e\left(\frac{4}{77}\right)$$ $$e\left(\frac{230}{231}\right)$$ $$e\left(\frac{155}{462}\right)$$ $$e\left(\frac{25}{66}\right)$$
$$\chi_{3381}(457,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{193}{231}\right)$$ $$e\left(\frac{155}{231}\right)$$ $$e\left(\frac{17}{462}\right)$$ $$e\left(\frac{39}{77}\right)$$ $$e\left(\frac{403}{462}\right)$$ $$e\left(\frac{43}{462}\right)$$ $$e\left(\frac{69}{77}\right)$$ $$e\left(\frac{79}{231}\right)$$ $$e\left(\frac{229}{462}\right)$$ $$e\left(\frac{5}{66}\right)$$
$$\chi_{3381}(550,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{218}{231}\right)$$ $$e\left(\frac{205}{231}\right)$$ $$e\left(\frac{97}{462}\right)$$ $$e\left(\frac{64}{77}\right)$$ $$e\left(\frac{71}{462}\right)$$ $$e\left(\frac{191}{462}\right)$$ $$e\left(\frac{54}{77}\right)$$ $$e\left(\frac{179}{231}\right)$$ $$e\left(\frac{437}{462}\right)$$ $$e\left(\frac{13}{66}\right)$$
$$\chi_{3381}(562,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{184}{231}\right)$$ $$e\left(\frac{137}{231}\right)$$ $$e\left(\frac{173}{462}\right)$$ $$e\left(\frac{30}{77}\right)$$ $$e\left(\frac{79}{462}\right)$$ $$e\left(\frac{193}{462}\right)$$ $$e\left(\frac{59}{77}\right)$$ $$e\left(\frac{43}{231}\right)$$ $$e\left(\frac{265}{462}\right)$$ $$e\left(\frac{47}{66}\right)$$
$$\chi_{3381}(571,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{194}{231}\right)$$ $$e\left(\frac{157}{231}\right)$$ $$e\left(\frac{205}{462}\right)$$ $$e\left(\frac{40}{77}\right)$$ $$e\left(\frac{131}{462}\right)$$ $$e\left(\frac{437}{462}\right)$$ $$e\left(\frac{53}{77}\right)$$ $$e\left(\frac{83}{231}\right)$$ $$e\left(\frac{71}{462}\right)$$ $$e\left(\frac{37}{66}\right)$$
$$\chi_{3381}(592,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{191}{231}\right)$$ $$e\left(\frac{151}{231}\right)$$ $$e\left(\frac{103}{462}\right)$$ $$e\left(\frac{37}{77}\right)$$ $$e\left(\frac{23}{462}\right)$$ $$e\left(\frac{179}{462}\right)$$ $$e\left(\frac{24}{77}\right)$$ $$e\left(\frac{71}{231}\right)$$ $$e\left(\frac{83}{462}\right)$$ $$e\left(\frac{7}{66}\right)$$
$$\chi_{3381}(613,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{104}{231}\right)$$ $$e\left(\frac{208}{231}\right)$$ $$e\left(\frac{379}{462}\right)$$ $$e\left(\frac{27}{77}\right)$$ $$e\left(\frac{125}{462}\right)$$ $$e\left(\frac{89}{462}\right)$$ $$e\left(\frac{30}{77}\right)$$ $$e\left(\frac{185}{231}\right)$$ $$e\left(\frac{431}{462}\right)$$ $$e\left(\frac{61}{66}\right)$$
$$\chi_{3381}(688,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{64}{231}\right)$$ $$e\left(\frac{128}{231}\right)$$ $$e\left(\frac{251}{462}\right)$$ $$e\left(\frac{64}{77}\right)$$ $$e\left(\frac{379}{462}\right)$$ $$e\left(\frac{37}{462}\right)$$ $$e\left(\frac{54}{77}\right)$$ $$e\left(\frac{25}{231}\right)$$ $$e\left(\frac{283}{462}\right)$$ $$e\left(\frac{35}{66}\right)$$
$$\chi_{3381}(697,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{113}{231}\right)$$ $$e\left(\frac{226}{231}\right)$$ $$e\left(\frac{223}{462}\right)$$ $$e\left(\frac{36}{77}\right)$$ $$e\left(\frac{449}{462}\right)$$ $$e\left(\frac{401}{462}\right)$$ $$e\left(\frac{40}{77}\right)$$ $$e\left(\frac{221}{231}\right)$$ $$e\left(\frac{395}{462}\right)$$ $$e\left(\frac{19}{66}\right)$$
$$\chi_{3381}(709,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{205}{231}\right)$$ $$e\left(\frac{179}{231}\right)$$ $$e\left(\frac{425}{462}\right)$$ $$e\left(\frac{51}{77}\right)$$ $$e\left(\frac{373}{462}\right)$$ $$e\left(\frac{151}{462}\right)$$ $$e\left(\frac{31}{77}\right)$$ $$e\left(\frac{127}{231}\right)$$ $$e\left(\frac{181}{462}\right)$$ $$e\left(\frac{59}{66}\right)$$
$$\chi_{3381}(718,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{131}{231}\right)$$ $$e\left(\frac{31}{231}\right)$$ $$e\left(\frac{373}{462}\right)$$ $$e\left(\frac{54}{77}\right)$$ $$e\left(\frac{173}{462}\right)$$ $$e\left(\frac{101}{462}\right)$$ $$e\left(\frac{60}{77}\right)$$ $$e\left(\frac{62}{231}\right)$$ $$e\left(\frac{323}{462}\right)$$ $$e\left(\frac{1}{66}\right)$$
$$\chi_{3381}(730,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{136}{231}\right)$$ $$e\left(\frac{41}{231}\right)$$ $$e\left(\frac{389}{462}\right)$$ $$e\left(\frac{59}{77}\right)$$ $$e\left(\frac{199}{462}\right)$$ $$e\left(\frac{223}{462}\right)$$ $$e\left(\frac{57}{77}\right)$$ $$e\left(\frac{82}{231}\right)$$ $$e\left(\frac{457}{462}\right)$$ $$e\left(\frac{29}{66}\right)$$
$$\chi_{3381}(751,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{214}{231}\right)$$ $$e\left(\frac{197}{231}\right)$$ $$e\left(\frac{269}{462}\right)$$ $$e\left(\frac{60}{77}\right)$$ $$e\left(\frac{235}{462}\right)$$ $$e\left(\frac{1}{462}\right)$$ $$e\left(\frac{41}{77}\right)$$ $$e\left(\frac{163}{231}\right)$$ $$e\left(\frac{145}{462}\right)$$ $$e\left(\frac{17}{66}\right)$$
$$\chi_{3381}(793,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{13}{231}\right)$$ $$e\left(\frac{26}{231}\right)$$ $$e\left(\frac{365}{462}\right)$$ $$e\left(\frac{13}{77}\right)$$ $$e\left(\frac{391}{462}\right)$$ $$e\left(\frac{271}{462}\right)$$ $$e\left(\frac{23}{77}\right)$$ $$e\left(\frac{52}{231}\right)$$ $$e\left(\frac{25}{462}\right)$$ $$e\left(\frac{53}{66}\right)$$
$$\chi_{3381}(835,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{190}{231}\right)$$ $$e\left(\frac{149}{231}\right)$$ $$e\left(\frac{377}{462}\right)$$ $$e\left(\frac{36}{77}\right)$$ $$e\left(\frac{295}{462}\right)$$ $$e\left(\frac{247}{462}\right)$$ $$e\left(\frac{40}{77}\right)$$ $$e\left(\frac{67}{231}\right)$$ $$e\left(\frac{241}{462}\right)$$ $$e\left(\frac{41}{66}\right)$$
$$\chi_{3381}(856,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{142}{231}\right)$$ $$e\left(\frac{53}{231}\right)$$ $$e\left(\frac{131}{462}\right)$$ $$e\left(\frac{65}{77}\right)$$ $$e\left(\frac{415}{462}\right)$$ $$e\left(\frac{277}{462}\right)$$ $$e\left(\frac{38}{77}\right)$$ $$e\left(\frac{106}{231}\right)$$ $$e\left(\frac{433}{462}\right)$$ $$e\left(\frac{23}{66}\right)$$
$$\chi_{3381}(865,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{89}{231}\right)$$ $$e\left(\frac{178}{231}\right)$$ $$e\left(\frac{331}{462}\right)$$ $$e\left(\frac{12}{77}\right)$$ $$e\left(\frac{47}{462}\right)$$ $$e\left(\frac{185}{462}\right)$$ $$e\left(\frac{39}{77}\right)$$ $$e\left(\frac{125}{231}\right)$$ $$e\left(\frac{29}{462}\right)$$ $$e\left(\frac{43}{66}\right)$$
$$\chi_{3381}(907,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{41}{231}\right)$$ $$e\left(\frac{82}{231}\right)$$ $$e\left(\frac{85}{462}\right)$$ $$e\left(\frac{41}{77}\right)$$ $$e\left(\frac{167}{462}\right)$$ $$e\left(\frac{215}{462}\right)$$ $$e\left(\frac{37}{77}\right)$$ $$e\left(\frac{164}{231}\right)$$ $$e\left(\frac{221}{462}\right)$$ $$e\left(\frac{25}{66}\right)$$