# Properties

 Label 837.cj Modulus $837$ Conductor $837$ Order $90$ Real no Primitive yes Minimal yes Parity odd

# Related objects

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

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

sage: M = H._module

sage: chi = DirichletCharacter(H, M([40,33]))

sage: chi.galois_orbit()

pari: [g,chi] = znchar(Mod(13,837))

pari: order = charorder(g,chi)

pari: [ 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: $$90$$ 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: $\Q(\zeta_{45})$ Fixed field: Number field defined by a degree 90 polynomial

## Characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$11$$ $$13$$ $$14$$ $$16$$
$$\chi_{837}(13,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{11}{45}\right)$$ $$e\left(\frac{22}{45}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{17}{45}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{19}{90}\right)$$ $$e\left(\frac{53}{90}\right)$$ $$e\left(\frac{28}{45}\right)$$ $$e\left(\frac{44}{45}\right)$$
$$\chi_{837}(22,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{17}{45}\right)$$ $$e\left(\frac{34}{45}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{14}{45}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{13}{90}\right)$$ $$e\left(\frac{41}{90}\right)$$ $$e\left(\frac{31}{45}\right)$$ $$e\left(\frac{23}{45}\right)$$
$$\chi_{837}(34,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{31}{45}\right)$$ $$e\left(\frac{17}{45}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{7}{45}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{29}{90}\right)$$ $$e\left(\frac{43}{90}\right)$$ $$e\left(\frac{38}{45}\right)$$ $$e\left(\frac{34}{45}\right)$$
$$\chi_{837}(43,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{19}{45}\right)$$ $$e\left(\frac{38}{45}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{13}{45}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{41}{90}\right)$$ $$e\left(\frac{67}{90}\right)$$ $$e\left(\frac{32}{45}\right)$$ $$e\left(\frac{31}{45}\right)$$
$$\chi_{837}(79,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{7}{45}\right)$$ $$e\left(\frac{14}{45}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{19}{45}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{53}{90}\right)$$ $$e\left(\frac{1}{90}\right)$$ $$e\left(\frac{26}{45}\right)$$ $$e\left(\frac{28}{45}\right)$$
$$\chi_{837}(166,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{23}{45}\right)$$ $$e\left(\frac{1}{45}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{11}{45}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{7}{90}\right)$$ $$e\left(\frac{29}{90}\right)$$ $$e\left(\frac{34}{45}\right)$$ $$e\left(\frac{2}{45}\right)$$
$$\chi_{837}(238,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{44}{45}\right)$$ $$e\left(\frac{43}{45}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{23}{45}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{31}{90}\right)$$ $$e\left(\frac{77}{90}\right)$$ $$e\left(\frac{22}{45}\right)$$ $$e\left(\frac{41}{45}\right)$$
$$\chi_{837}(241,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{43}{45}\right)$$ $$e\left(\frac{41}{45}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{1}{45}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{17}{90}\right)$$ $$e\left(\frac{19}{90}\right)$$ $$e\left(\frac{44}{45}\right)$$ $$e\left(\frac{37}{45}\right)$$
$$\chi_{837}(292,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{26}{45}\right)$$ $$e\left(\frac{7}{45}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{32}{45}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{49}{90}\right)$$ $$e\left(\frac{23}{90}\right)$$ $$e\left(\frac{13}{45}\right)$$ $$e\left(\frac{14}{45}\right)$$
$$\chi_{837}(301,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{32}{45}\right)$$ $$e\left(\frac{19}{45}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{29}{45}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{43}{90}\right)$$ $$e\left(\frac{11}{90}\right)$$ $$e\left(\frac{16}{45}\right)$$ $$e\left(\frac{38}{45}\right)$$
$$\chi_{837}(313,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{45}\right)$$ $$e\left(\frac{2}{45}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{22}{45}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{59}{90}\right)$$ $$e\left(\frac{13}{90}\right)$$ $$e\left(\frac{23}{45}\right)$$ $$e\left(\frac{4}{45}\right)$$
$$\chi_{837}(322,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{34}{45}\right)$$ $$e\left(\frac{23}{45}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{28}{45}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{71}{90}\right)$$ $$e\left(\frac{37}{90}\right)$$ $$e\left(\frac{17}{45}\right)$$ $$e\left(\frac{1}{45}\right)$$
$$\chi_{837}(358,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{22}{45}\right)$$ $$e\left(\frac{44}{45}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{34}{45}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{83}{90}\right)$$ $$e\left(\frac{61}{90}\right)$$ $$e\left(\frac{11}{45}\right)$$ $$e\left(\frac{43}{45}\right)$$
$$\chi_{837}(445,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{38}{45}\right)$$ $$e\left(\frac{31}{45}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{26}{45}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{37}{90}\right)$$ $$e\left(\frac{89}{90}\right)$$ $$e\left(\frac{19}{45}\right)$$ $$e\left(\frac{17}{45}\right)$$
$$\chi_{837}(517,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{14}{45}\right)$$ $$e\left(\frac{28}{45}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{38}{45}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{61}{90}\right)$$ $$e\left(\frac{47}{90}\right)$$ $$e\left(\frac{7}{45}\right)$$ $$e\left(\frac{11}{45}\right)$$
$$\chi_{837}(520,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{13}{45}\right)$$ $$e\left(\frac{26}{45}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{16}{45}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{47}{90}\right)$$ $$e\left(\frac{79}{90}\right)$$ $$e\left(\frac{29}{45}\right)$$ $$e\left(\frac{7}{45}\right)$$
$$\chi_{837}(571,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{41}{45}\right)$$ $$e\left(\frac{37}{45}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{2}{45}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{79}{90}\right)$$ $$e\left(\frac{83}{90}\right)$$ $$e\left(\frac{43}{45}\right)$$ $$e\left(\frac{29}{45}\right)$$
$$\chi_{837}(580,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{2}{45}\right)$$ $$e\left(\frac{4}{45}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{44}{45}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{73}{90}\right)$$ $$e\left(\frac{71}{90}\right)$$ $$e\left(\frac{1}{45}\right)$$ $$e\left(\frac{8}{45}\right)$$
$$\chi_{837}(592,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{16}{45}\right)$$ $$e\left(\frac{32}{45}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{37}{45}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{89}{90}\right)$$ $$e\left(\frac{73}{90}\right)$$ $$e\left(\frac{8}{45}\right)$$ $$e\left(\frac{19}{45}\right)$$
$$\chi_{837}(601,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{4}{45}\right)$$ $$e\left(\frac{8}{45}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{43}{45}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{11}{90}\right)$$ $$e\left(\frac{7}{90}\right)$$ $$e\left(\frac{2}{45}\right)$$ $$e\left(\frac{16}{45}\right)$$
$$\chi_{837}(637,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{37}{45}\right)$$ $$e\left(\frac{29}{45}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{4}{45}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{23}{90}\right)$$ $$e\left(\frac{31}{90}\right)$$ $$e\left(\frac{41}{45}\right)$$ $$e\left(\frac{13}{45}\right)$$
$$\chi_{837}(724,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{8}{45}\right)$$ $$e\left(\frac{16}{45}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{41}{45}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{67}{90}\right)$$ $$e\left(\frac{59}{90}\right)$$ $$e\left(\frac{4}{45}\right)$$ $$e\left(\frac{32}{45}\right)$$
$$\chi_{837}(796,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{29}{45}\right)$$ $$e\left(\frac{13}{45}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{8}{45}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{1}{90}\right)$$ $$e\left(\frac{17}{90}\right)$$ $$e\left(\frac{37}{45}\right)$$ $$e\left(\frac{26}{45}\right)$$
$$\chi_{837}(799,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{28}{45}\right)$$ $$e\left(\frac{11}{45}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{31}{45}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{77}{90}\right)$$ $$e\left(\frac{49}{90}\right)$$ $$e\left(\frac{14}{45}\right)$$ $$e\left(\frac{22}{45}\right)$$