# Properties

 Label 1045.co Modulus $1045$ Conductor $209$ Order $90$ 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(1045, base_ring=CyclotomicField(90))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,36,35]))

sage: chi.galois_orbit()

pari: [g,chi] = znchar(Mod(71,1045))

pari: order = charorder(g,chi)

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

## Basic properties

 Modulus: $$1045$$ Conductor: $$209$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$90$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from 209.x 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 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$6$$ $$7$$ $$8$$ $$9$$ $$12$$ $$13$$ $$14$$
$$\chi_{1045}(71,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{71}{90}\right)$$ $$e\left(\frac{23}{90}\right)$$ $$e\left(\frac{26}{45}\right)$$ $$e\left(\frac{2}{45}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{23}{45}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{31}{90}\right)$$ $$e\left(\frac{83}{90}\right)$$
$$\chi_{1045}(86,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{49}{90}\right)$$ $$e\left(\frac{7}{90}\right)$$ $$e\left(\frac{4}{45}\right)$$ $$e\left(\frac{28}{45}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{7}{45}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{29}{90}\right)$$ $$e\left(\frac{37}{90}\right)$$
$$\chi_{1045}(91,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{37}{90}\right)$$ $$e\left(\frac{31}{90}\right)$$ $$e\left(\frac{37}{45}\right)$$ $$e\left(\frac{34}{45}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{31}{45}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{77}{90}\right)$$ $$e\left(\frac{61}{90}\right)$$
$$\chi_{1045}(136,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{83}{90}\right)$$ $$e\left(\frac{89}{90}\right)$$ $$e\left(\frac{38}{45}\right)$$ $$e\left(\frac{41}{45}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{44}{45}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{73}{90}\right)$$ $$e\left(\frac{59}{90}\right)$$
$$\chi_{1045}(146,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{7}{90}\right)$$ $$e\left(\frac{1}{90}\right)$$ $$e\left(\frac{7}{45}\right)$$ $$e\left(\frac{4}{45}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{1}{45}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{17}{90}\right)$$ $$e\left(\frac{31}{90}\right)$$
$$\chi_{1045}(181,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{31}{90}\right)$$ $$e\left(\frac{43}{90}\right)$$ $$e\left(\frac{31}{45}\right)$$ $$e\left(\frac{37}{45}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{43}{45}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{11}{90}\right)$$ $$e\left(\frac{73}{90}\right)$$
$$\chi_{1045}(306,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{59}{90}\right)$$ $$e\left(\frac{47}{90}\right)$$ $$e\left(\frac{14}{45}\right)$$ $$e\left(\frac{8}{45}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{2}{45}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{79}{90}\right)$$ $$e\left(\frac{17}{90}\right)$$
$$\chi_{1045}(356,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{53}{90}\right)$$ $$e\left(\frac{59}{90}\right)$$ $$e\left(\frac{8}{45}\right)$$ $$e\left(\frac{11}{45}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{14}{45}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{13}{90}\right)$$ $$e\left(\frac{29}{90}\right)$$
$$\chi_{1045}(401,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{41}{90}\right)$$ $$e\left(\frac{83}{90}\right)$$ $$e\left(\frac{41}{45}\right)$$ $$e\left(\frac{17}{45}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{38}{45}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{61}{90}\right)$$ $$e\left(\frac{53}{90}\right)$$
$$\chi_{1045}(421,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{47}{90}\right)$$ $$e\left(\frac{71}{90}\right)$$ $$e\left(\frac{2}{45}\right)$$ $$e\left(\frac{14}{45}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{26}{45}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{37}{90}\right)$$ $$e\left(\frac{41}{90}\right)$$
$$\chi_{1045}(466,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{13}{90}\right)$$ $$e\left(\frac{79}{90}\right)$$ $$e\left(\frac{13}{45}\right)$$ $$e\left(\frac{1}{45}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{34}{45}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{83}{90}\right)$$ $$e\left(\frac{19}{90}\right)$$
$$\chi_{1045}(471,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{19}{90}\right)$$ $$e\left(\frac{67}{90}\right)$$ $$e\left(\frac{19}{45}\right)$$ $$e\left(\frac{43}{45}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{22}{45}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{59}{90}\right)$$ $$e\left(\frac{7}{90}\right)$$
$$\chi_{1045}(526,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{79}{90}\right)$$ $$e\left(\frac{37}{90}\right)$$ $$e\left(\frac{34}{45}\right)$$ $$e\left(\frac{13}{45}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{37}{45}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{89}{90}\right)$$ $$e\left(\frac{67}{90}\right)$$
$$\chi_{1045}(566,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{90}\right)$$ $$e\left(\frac{13}{90}\right)$$ $$e\left(\frac{1}{45}\right)$$ $$e\left(\frac{7}{45}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{13}{45}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{41}{90}\right)$$ $$e\left(\frac{43}{90}\right)$$
$$\chi_{1045}(621,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{61}{90}\right)$$ $$e\left(\frac{73}{90}\right)$$ $$e\left(\frac{16}{45}\right)$$ $$e\left(\frac{22}{45}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{28}{45}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{71}{90}\right)$$ $$e\left(\frac{13}{90}\right)$$
$$\chi_{1045}(641,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{17}{90}\right)$$ $$e\left(\frac{41}{90}\right)$$ $$e\left(\frac{17}{45}\right)$$ $$e\left(\frac{29}{45}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{41}{45}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{67}{90}\right)$$ $$e\left(\frac{11}{90}\right)$$
$$\chi_{1045}(686,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{23}{90}\right)$$ $$e\left(\frac{29}{90}\right)$$ $$e\left(\frac{23}{45}\right)$$ $$e\left(\frac{26}{45}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{29}{45}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{43}{90}\right)$$ $$e\left(\frac{89}{90}\right)$$
$$\chi_{1045}(751,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{67}{90}\right)$$ $$e\left(\frac{61}{90}\right)$$ $$e\left(\frac{22}{45}\right)$$ $$e\left(\frac{19}{45}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{16}{45}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{47}{90}\right)$$ $$e\left(\frac{1}{90}\right)$$
$$\chi_{1045}(801,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{29}{90}\right)$$ $$e\left(\frac{17}{90}\right)$$ $$e\left(\frac{29}{45}\right)$$ $$e\left(\frac{23}{45}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{17}{45}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{19}{90}\right)$$ $$e\left(\frac{77}{90}\right)$$
$$\chi_{1045}(851,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{73}{90}\right)$$ $$e\left(\frac{49}{90}\right)$$ $$e\left(\frac{28}{45}\right)$$ $$e\left(\frac{16}{45}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{4}{45}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{23}{90}\right)$$ $$e\left(\frac{79}{90}\right)$$
$$\chi_{1045}(896,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{11}{90}\right)$$ $$e\left(\frac{53}{90}\right)$$ $$e\left(\frac{11}{45}\right)$$ $$e\left(\frac{32}{45}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{8}{45}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{90}\right)$$ $$e\left(\frac{23}{90}\right)$$
$$\chi_{1045}(906,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{43}{90}\right)$$ $$e\left(\frac{19}{90}\right)$$ $$e\left(\frac{43}{45}\right)$$ $$e\left(\frac{31}{45}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{19}{45}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{53}{90}\right)$$ $$e\left(\frac{49}{90}\right)$$
$$\chi_{1045}(971,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{77}{90}\right)$$ $$e\left(\frac{11}{90}\right)$$ $$e\left(\frac{32}{45}\right)$$ $$e\left(\frac{44}{45}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{11}{45}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{7}{90}\right)$$ $$e\left(\frac{71}{90}\right)$$
$$\chi_{1045}(1021,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{89}{90}\right)$$ $$e\left(\frac{77}{90}\right)$$ $$e\left(\frac{44}{45}\right)$$ $$e\left(\frac{38}{45}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{32}{45}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{49}{90}\right)$$ $$e\left(\frac{47}{90}\right)$$