# Properties

 Label 8470.cq Modulus $8470$ Conductor $4235$ Order $66$ Real no Primitive no Minimal yes Parity even

# Related objects

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

sage: H = DirichletGroup(8470, base_ring=CyclotomicField(66))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([33,44,18]))

sage: chi.galois_orbit()

pari: [g,chi] = znchar(Mod(529,8470))

pari: order = charorder(g,chi)

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

## Basic properties

 Modulus: $$8470$$ Conductor: $$4235$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$66$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from 4235.cn 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: $$\Q(\zeta_{33})$$ Fixed field: Number field defined by a degree 66 polynomial

## Characters in Galois orbit

Character $$-1$$ $$1$$ $$3$$ $$9$$ $$13$$ $$17$$ $$19$$ $$23$$ $$27$$ $$29$$ $$31$$ $$37$$
$$\chi_{8470}(529,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{35}{66}\right)$$ $$e\left(\frac{32}{33}\right)$$ $$e\left(\frac{61}{66}\right)$$ $$-1$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{4}{33}\right)$$ $$e\left(\frac{19}{66}\right)$$
$$\chi_{8470}(639,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{7}{66}\right)$$ $$e\left(\frac{13}{33}\right)$$ $$e\left(\frac{65}{66}\right)$$ $$-1$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{14}{33}\right)$$ $$e\left(\frac{17}{66}\right)$$
$$\chi_{8470}(1299,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{59}{66}\right)$$ $$e\left(\frac{20}{33}\right)$$ $$e\left(\frac{1}{66}\right)$$ $$-1$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{19}{33}\right)$$ $$e\left(\frac{49}{66}\right)$$
$$\chi_{8470}(1409,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{31}{66}\right)$$ $$e\left(\frac{1}{33}\right)$$ $$e\left(\frac{5}{66}\right)$$ $$-1$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{29}{33}\right)$$ $$e\left(\frac{47}{66}\right)$$
$$\chi_{8470}(2069,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{17}{66}\right)$$ $$e\left(\frac{8}{33}\right)$$ $$e\left(\frac{7}{66}\right)$$ $$-1$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{1}{33}\right)$$ $$e\left(\frac{13}{66}\right)$$
$$\chi_{8470}(2839,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{41}{66}\right)$$ $$e\left(\frac{29}{33}\right)$$ $$e\left(\frac{13}{66}\right)$$ $$-1$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{16}{33}\right)$$ $$e\left(\frac{43}{66}\right)$$
$$\chi_{8470}(2949,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{13}{66}\right)$$ $$e\left(\frac{10}{33}\right)$$ $$e\left(\frac{17}{66}\right)$$ $$-1$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{26}{33}\right)$$ $$e\left(\frac{41}{66}\right)$$
$$\chi_{8470}(3609,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{65}{66}\right)$$ $$e\left(\frac{17}{33}\right)$$ $$e\left(\frac{19}{66}\right)$$ $$-1$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{31}{33}\right)$$ $$e\left(\frac{7}{66}\right)$$
$$\chi_{8470}(3719,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{37}{66}\right)$$ $$e\left(\frac{31}{33}\right)$$ $$e\left(\frac{23}{66}\right)$$ $$-1$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{8}{33}\right)$$ $$e\left(\frac{5}{66}\right)$$
$$\chi_{8470}(4379,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{23}{66}\right)$$ $$e\left(\frac{5}{33}\right)$$ $$e\left(\frac{25}{66}\right)$$ $$-1$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{13}{33}\right)$$ $$e\left(\frac{37}{66}\right)$$
$$\chi_{8470}(4489,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{61}{66}\right)$$ $$e\left(\frac{19}{33}\right)$$ $$e\left(\frac{29}{66}\right)$$ $$-1$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{23}{33}\right)$$ $$e\left(\frac{35}{66}\right)$$
$$\chi_{8470}(5149,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{47}{66}\right)$$ $$e\left(\frac{26}{33}\right)$$ $$e\left(\frac{31}{66}\right)$$ $$-1$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{28}{33}\right)$$ $$e\left(\frac{1}{66}\right)$$
$$\chi_{8470}(5259,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{19}{66}\right)$$ $$e\left(\frac{7}{33}\right)$$ $$e\left(\frac{35}{66}\right)$$ $$-1$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{5}{33}\right)$$ $$e\left(\frac{65}{66}\right)$$
$$\chi_{8470}(5919,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{5}{66}\right)$$ $$e\left(\frac{14}{33}\right)$$ $$e\left(\frac{37}{66}\right)$$ $$-1$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{10}{33}\right)$$ $$e\left(\frac{31}{66}\right)$$
$$\chi_{8470}(6029,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{43}{66}\right)$$ $$e\left(\frac{28}{33}\right)$$ $$e\left(\frac{41}{66}\right)$$ $$-1$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{20}{33}\right)$$ $$e\left(\frac{29}{66}\right)$$
$$\chi_{8470}(6689,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{29}{66}\right)$$ $$e\left(\frac{2}{33}\right)$$ $$e\left(\frac{43}{66}\right)$$ $$-1$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{25}{33}\right)$$ $$e\left(\frac{61}{66}\right)$$
$$\chi_{8470}(6799,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{17}{22}\right)$$ $$e\left(\frac{1}{66}\right)$$ $$e\left(\frac{16}{33}\right)$$ $$e\left(\frac{47}{66}\right)$$ $$-1$$ $$e\left(\frac{9}{11}\right)$$ $$e\left(\frac{2}{33}\right)$$ $$e\left(\frac{59}{66}\right)$$
$$\chi_{8470}(7459,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{21}{22}\right)$$ $$e\left(\frac{53}{66}\right)$$ $$e\left(\frac{23}{33}\right)$$ $$e\left(\frac{49}{66}\right)$$ $$-1$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{7}{33}\right)$$ $$e\left(\frac{25}{66}\right)$$
$$\chi_{8470}(7569,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{25}{66}\right)$$ $$e\left(\frac{4}{33}\right)$$ $$e\left(\frac{53}{66}\right)$$ $$-1$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{17}{33}\right)$$ $$e\left(\frac{23}{66}\right)$$
$$\chi_{8470}(8339,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{19}{22}\right)$$ $$e\left(\frac{49}{66}\right)$$ $$e\left(\frac{25}{33}\right)$$ $$e\left(\frac{59}{66}\right)$$ $$-1$$ $$e\left(\frac{1}{11}\right)$$ $$e\left(\frac{32}{33}\right)$$ $$e\left(\frac{53}{66}\right)$$