# Properties

 Label 837.cb Modulus $837$ Conductor $837$ Order $45$ Real no Primitive yes Minimal yes Parity even

# Related objects

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

H = DirichletGroup(837, base_ring=CyclotomicField(90))

M = H._module

chi = DirichletCharacter(H, M([70,66]))

chi.galois_orbit()

[g,chi] = znchar(Mod(76,837))

order = charorder(g,chi)

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

## Basic properties

 Modulus: $$837$$ Conductor: $$837$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$45$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: yes 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: $\Q(\zeta_{45})$ Fixed field: Number field defined by a degree 45 polynomial

## Characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$11$$ $$13$$ $$14$$ $$16$$
$$\chi_{837}(76,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{17}{45}\right)$$ $$e\left(\frac{34}{45}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{44}{45}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{44}{45}\right)$$ $$e\left(\frac{13}{45}\right)$$ $$e\left(\frac{16}{45}\right)$$ $$e\left(\frac{23}{45}\right)$$
$$\chi_{837}(112,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{14}{45}\right)$$ $$e\left(\frac{28}{45}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{23}{45}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{23}{45}\right)$$ $$e\left(\frac{16}{45}\right)$$ $$e\left(\frac{37}{45}\right)$$ $$e\left(\frac{11}{45}\right)$$
$$\chi_{837}(121,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{11}{45}\right)$$ $$e\left(\frac{22}{45}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{2}{45}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{2}{45}\right)$$ $$e\left(\frac{19}{45}\right)$$ $$e\left(\frac{13}{45}\right)$$ $$e\left(\frac{44}{45}\right)$$
$$\chi_{837}(133,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{7}{45}\right)$$ $$e\left(\frac{14}{45}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{34}{45}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{34}{45}\right)$$ $$e\left(\frac{8}{45}\right)$$ $$e\left(\frac{41}{45}\right)$$ $$e\left(\frac{28}{45}\right)$$
$$\chi_{837}(142,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{31}{45}\right)$$ $$e\left(\frac{17}{45}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{22}{45}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{22}{45}\right)$$ $$e\left(\frac{29}{45}\right)$$ $$e\left(\frac{8}{45}\right)$$ $$e\left(\frac{34}{45}\right)$$
$$\chi_{837}(193,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{23}{45}\right)$$ $$e\left(\frac{1}{45}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{41}{45}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{41}{45}\right)$$ $$e\left(\frac{7}{45}\right)$$ $$e\left(\frac{19}{45}\right)$$ $$e\left(\frac{2}{45}\right)$$
$$\chi_{837}(196,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{4}{45}\right)$$ $$e\left(\frac{8}{45}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{13}{45}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{13}{45}\right)$$ $$e\left(\frac{11}{45}\right)$$ $$e\left(\frac{17}{45}\right)$$ $$e\left(\frac{16}{45}\right)$$
$$\chi_{837}(268,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{43}{45}\right)$$ $$e\left(\frac{41}{45}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{16}{45}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{16}{45}\right)$$ $$e\left(\frac{17}{45}\right)$$ $$e\left(\frac{14}{45}\right)$$ $$e\left(\frac{37}{45}\right)$$
$$\chi_{837}(355,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{32}{45}\right)$$ $$e\left(\frac{19}{45}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{14}{45}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{14}{45}\right)$$ $$e\left(\frac{43}{45}\right)$$ $$e\left(\frac{1}{45}\right)$$ $$e\left(\frac{38}{45}\right)$$
$$\chi_{837}(391,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{29}{45}\right)$$ $$e\left(\frac{13}{45}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{38}{45}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{38}{45}\right)$$ $$e\left(\frac{1}{45}\right)$$ $$e\left(\frac{22}{45}\right)$$ $$e\left(\frac{26}{45}\right)$$
$$\chi_{837}(400,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{26}{45}\right)$$ $$e\left(\frac{7}{45}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{17}{45}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{17}{45}\right)$$ $$e\left(\frac{4}{45}\right)$$ $$e\left(\frac{43}{45}\right)$$ $$e\left(\frac{14}{45}\right)$$
$$\chi_{837}(412,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{22}{45}\right)$$ $$e\left(\frac{44}{45}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{4}{45}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{4}{45}\right)$$ $$e\left(\frac{38}{45}\right)$$ $$e\left(\frac{26}{45}\right)$$ $$e\left(\frac{43}{45}\right)$$
$$\chi_{837}(421,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{45}\right)$$ $$e\left(\frac{2}{45}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{37}{45}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{37}{45}\right)$$ $$e\left(\frac{14}{45}\right)$$ $$e\left(\frac{38}{45}\right)$$ $$e\left(\frac{4}{45}\right)$$
$$\chi_{837}(472,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{38}{45}\right)$$ $$e\left(\frac{31}{45}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{11}{45}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{11}{45}\right)$$ $$e\left(\frac{37}{45}\right)$$ $$e\left(\frac{4}{45}\right)$$ $$e\left(\frac{17}{45}\right)$$
$$\chi_{837}(475,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{19}{45}\right)$$ $$e\left(\frac{38}{45}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{28}{45}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{28}{45}\right)$$ $$e\left(\frac{41}{45}\right)$$ $$e\left(\frac{2}{45}\right)$$ $$e\left(\frac{31}{45}\right)$$
$$\chi_{837}(547,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{13}{45}\right)$$ $$e\left(\frac{26}{45}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{31}{45}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{31}{45}\right)$$ $$e\left(\frac{2}{45}\right)$$ $$e\left(\frac{44}{45}\right)$$ $$e\left(\frac{7}{45}\right)$$
$$\chi_{837}(634,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{2}{45}\right)$$ $$e\left(\frac{4}{45}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{29}{45}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{29}{45}\right)$$ $$e\left(\frac{28}{45}\right)$$ $$e\left(\frac{31}{45}\right)$$ $$e\left(\frac{8}{45}\right)$$
$$\chi_{837}(670,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{44}{45}\right)$$ $$e\left(\frac{43}{45}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{8}{45}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{8}{45}\right)$$ $$e\left(\frac{31}{45}\right)$$ $$e\left(\frac{7}{45}\right)$$ $$e\left(\frac{41}{45}\right)$$
$$\chi_{837}(679,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{41}{45}\right)$$ $$e\left(\frac{37}{45}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{32}{45}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{32}{45}\right)$$ $$e\left(\frac{34}{45}\right)$$ $$e\left(\frac{28}{45}\right)$$ $$e\left(\frac{29}{45}\right)$$
$$\chi_{837}(691,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{37}{45}\right)$$ $$e\left(\frac{29}{45}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{19}{45}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{19}{45}\right)$$ $$e\left(\frac{23}{45}\right)$$ $$e\left(\frac{11}{45}\right)$$ $$e\left(\frac{13}{45}\right)$$
$$\chi_{837}(700,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{16}{45}\right)$$ $$e\left(\frac{32}{45}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{7}{45}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{7}{45}\right)$$ $$e\left(\frac{44}{45}\right)$$ $$e\left(\frac{23}{45}\right)$$ $$e\left(\frac{19}{45}\right)$$
$$\chi_{837}(751,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{8}{45}\right)$$ $$e\left(\frac{16}{45}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{26}{45}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{26}{45}\right)$$ $$e\left(\frac{22}{45}\right)$$ $$e\left(\frac{34}{45}\right)$$ $$e\left(\frac{32}{45}\right)$$
$$\chi_{837}(754,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{34}{45}\right)$$ $$e\left(\frac{23}{45}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{43}{45}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{43}{45}\right)$$ $$e\left(\frac{26}{45}\right)$$ $$e\left(\frac{32}{45}\right)$$ $$e\left(\frac{1}{45}\right)$$
$$\chi_{837}(826,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{28}{45}\right)$$ $$e\left(\frac{11}{45}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{1}{45}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{1}{45}\right)$$ $$e\left(\frac{32}{45}\right)$$ $$e\left(\frac{29}{45}\right)$$ $$e\left(\frac{22}{45}\right)$$