# Properties

 Label 3381.bu Modulus $3381$ Conductor $161$ Order $66$ Real no Primitive no Minimal no Parity odd

# Related objects

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

sage: H = DirichletGroup(3381, base_ring=CyclotomicField(66))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,11,18]))

sage: chi.galois_orbit()

pari: [g,chi] = znchar(Mod(31,3381))

pari: order = charorder(g,chi)

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

## Basic properties

 Modulus: $$3381$$ Conductor: $$161$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$66$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from 161.n 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: $$\Q(\zeta_{33})$$ Fixed field: Number field defined by a degree 66 polynomial

## Characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$8$$ $$10$$ $$11$$ $$13$$ $$16$$ $$17$$ $$19$$
$$\chi_{3381}(31,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{29}{33}\right)$$ $$e\left(\frac{25}{33}\right)$$ $$e\left(\frac{7}{66}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{65}{66}\right)$$ $$e\left(\frac{4}{33}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{17}{33}\right)$$ $$e\left(\frac{5}{66}\right)$$ $$e\left(\frac{61}{66}\right)$$
$$\chi_{3381}(325,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{26}{33}\right)$$ $$e\left(\frac{19}{33}\right)$$ $$e\left(\frac{37}{66}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{23}{66}\right)$$ $$e\left(\frac{7}{33}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{5}{33}\right)$$ $$e\left(\frac{17}{66}\right)$$ $$e\left(\frac{49}{66}\right)$$
$$\chi_{3381}(472,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{5}{33}\right)$$ $$e\left(\frac{10}{33}\right)$$ $$e\left(\frac{49}{66}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{59}{66}\right)$$ $$e\left(\frac{28}{33}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{20}{33}\right)$$ $$e\left(\frac{35}{66}\right)$$ $$e\left(\frac{31}{66}\right)$$
$$\chi_{3381}(607,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{19}{33}\right)$$ $$e\left(\frac{5}{33}\right)$$ $$e\left(\frac{41}{66}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{13}{66}\right)$$ $$e\left(\frac{14}{33}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{10}{33}\right)$$ $$e\left(\frac{1}{66}\right)$$ $$e\left(\frac{65}{66}\right)$$
$$\chi_{3381}(754,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{25}{33}\right)$$ $$e\left(\frac{17}{33}\right)$$ $$e\left(\frac{47}{66}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{31}{66}\right)$$ $$e\left(\frac{8}{33}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{1}{33}\right)$$ $$e\left(\frac{43}{66}\right)$$ $$e\left(\frac{23}{66}\right)$$
$$\chi_{3381}(901,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{33}\right)$$ $$e\left(\frac{2}{33}\right)$$ $$e\left(\frac{23}{66}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{25}{66}\right)$$ $$e\left(\frac{32}{33}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{4}{33}\right)$$ $$e\left(\frac{7}{66}\right)$$ $$e\left(\frac{59}{66}\right)$$
$$\chi_{3381}(913,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{2}{33}\right)$$ $$e\left(\frac{4}{33}\right)$$ $$e\left(\frac{13}{66}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{17}{66}\right)$$ $$e\left(\frac{31}{33}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{8}{33}\right)$$ $$e\left(\frac{47}{66}\right)$$ $$e\left(\frac{19}{66}\right)$$
$$\chi_{3381}(1048,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{31}{33}\right)$$ $$e\left(\frac{29}{33}\right)$$ $$e\left(\frac{53}{66}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{49}{66}\right)$$ $$e\left(\frac{2}{33}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{25}{33}\right)$$ $$e\left(\frac{19}{66}\right)$$ $$e\left(\frac{47}{66}\right)$$
$$\chi_{3381}(1060,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{17}{33}\right)$$ $$e\left(\frac{1}{33}\right)$$ $$e\left(\frac{61}{66}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{29}{66}\right)$$ $$e\left(\frac{16}{33}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{2}{33}\right)$$ $$e\left(\frac{53}{66}\right)$$ $$e\left(\frac{13}{66}\right)$$
$$\chi_{3381}(1342,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{7}{33}\right)$$ $$e\left(\frac{14}{33}\right)$$ $$e\left(\frac{29}{66}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{43}{66}\right)$$ $$e\left(\frac{26}{33}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{28}{33}\right)$$ $$e\left(\frac{49}{66}\right)$$ $$e\left(\frac{17}{66}\right)$$
$$\chi_{3381}(1501,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{32}{33}\right)$$ $$e\left(\frac{31}{33}\right)$$ $$e\left(\frac{43}{66}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{41}{66}\right)$$ $$e\left(\frac{1}{33}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{29}{33}\right)$$ $$e\left(\frac{59}{66}\right)$$ $$e\left(\frac{7}{66}\right)$$
$$\chi_{3381}(1636,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{4}{33}\right)$$ $$e\left(\frac{8}{33}\right)$$ $$e\left(\frac{59}{66}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{1}{66}\right)$$ $$e\left(\frac{29}{33}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{16}{33}\right)$$ $$e\left(\frac{61}{66}\right)$$ $$e\left(\frac{5}{66}\right)$$
$$\chi_{3381}(1783,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{16}{33}\right)$$ $$e\left(\frac{32}{33}\right)$$ $$e\left(\frac{5}{66}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{37}{66}\right)$$ $$e\left(\frac{17}{33}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{31}{33}\right)$$ $$e\left(\frac{13}{66}\right)$$ $$e\left(\frac{53}{66}\right)$$
$$\chi_{3381}(2224,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{13}{33}\right)$$ $$e\left(\frac{26}{33}\right)$$ $$e\left(\frac{35}{66}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{61}{66}\right)$$ $$e\left(\frac{20}{33}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{19}{33}\right)$$ $$e\left(\frac{25}{66}\right)$$ $$e\left(\frac{41}{66}\right)$$
$$\chi_{3381}(2371,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{28}{33}\right)$$ $$e\left(\frac{23}{33}\right)$$ $$e\left(\frac{17}{66}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{7}{66}\right)$$ $$e\left(\frac{5}{33}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{13}{33}\right)$$ $$e\left(\frac{31}{66}\right)$$ $$e\left(\frac{35}{66}\right)$$
$$\chi_{3381}(2677,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{8}{33}\right)$$ $$e\left(\frac{16}{33}\right)$$ $$e\left(\frac{19}{66}\right)$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{35}{66}\right)$$ $$e\left(\frac{25}{33}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{32}{33}\right)$$ $$e\left(\frac{23}{66}\right)$$ $$e\left(\frac{43}{66}\right)$$
$$\chi_{3381}(2812,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{10}{33}\right)$$ $$e\left(\frac{20}{33}\right)$$ $$e\left(\frac{65}{66}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{19}{66}\right)$$ $$e\left(\frac{23}{33}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{7}{33}\right)$$ $$e\left(\frac{37}{66}\right)$$ $$e\left(\frac{29}{66}\right)$$
$$\chi_{3381}(2824,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{14}{33}\right)$$ $$e\left(\frac{28}{33}\right)$$ $$e\left(\frac{25}{66}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{53}{66}\right)$$ $$e\left(\frac{19}{33}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{23}{33}\right)$$ $$e\left(\frac{65}{66}\right)$$ $$e\left(\frac{1}{66}\right)$$
$$\chi_{3381}(2971,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{23}{33}\right)$$ $$e\left(\frac{13}{33}\right)$$ $$e\left(\frac{1}{66}\right)$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{47}{66}\right)$$ $$e\left(\frac{10}{33}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{26}{33}\right)$$ $$e\left(\frac{29}{66}\right)$$ $$e\left(\frac{37}{66}\right)$$
$$\chi_{3381}(3118,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{20}{33}\right)$$ $$e\left(\frac{7}{33}\right)$$ $$e\left(\frac{31}{66}\right)$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{5}{66}\right)$$ $$e\left(\frac{13}{33}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{14}{33}\right)$$ $$e\left(\frac{41}{66}\right)$$ $$e\left(\frac{25}{66}\right)$$