# Properties

 Label 1148.ck Modulus $1148$ Conductor $1148$ Order $120$ 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(1148, base_ring=CyclotomicField(120))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([60,100,27]))

sage: chi.galois_orbit()

pari: [g,chi] = znchar(Mod(19,1148))

pari: order = charorder(g,chi)

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

## Basic properties

 Modulus: $$1148$$ Conductor: $$1148$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$120$$ 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: $\Q(\zeta_{120})$ Fixed field: Number field defined by a degree 120 polynomial (not computed)

## First 31 of 32 characters in Galois orbit

Character $$-1$$ $$1$$ $$3$$ $$5$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$19$$ $$23$$ $$25$$
$$\chi_{1148}(19,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{7}{60}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{61}{120}\right)$$ $$e\left(\frac{19}{40}\right)$$ $$e\left(\frac{33}{40}\right)$$ $$e\left(\frac{31}{120}\right)$$ $$e\left(\frac{83}{120}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{7}{30}\right)$$
$$\chi_{1148}(47,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{43}{60}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{109}{120}\right)$$ $$e\left(\frac{11}{40}\right)$$ $$e\left(\frac{17}{40}\right)$$ $$e\left(\frac{79}{120}\right)$$ $$e\left(\frac{107}{120}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{13}{30}\right)$$
$$\chi_{1148}(75,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{37}{60}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{31}{120}\right)$$ $$e\left(\frac{9}{40}\right)$$ $$e\left(\frac{3}{40}\right)$$ $$e\left(\frac{61}{120}\right)$$ $$e\left(\frac{113}{120}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{7}{30}\right)$$
$$\chi_{1148}(171,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{17}{60}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{11}{120}\right)$$ $$e\left(\frac{29}{40}\right)$$ $$e\left(\frac{23}{40}\right)$$ $$e\left(\frac{41}{120}\right)$$ $$e\left(\frac{13}{120}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{17}{30}\right)$$
$$\chi_{1148}(199,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{23}{60}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{89}{120}\right)$$ $$e\left(\frac{31}{40}\right)$$ $$e\left(\frac{37}{40}\right)$$ $$e\left(\frac{59}{120}\right)$$ $$e\left(\frac{7}{120}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{23}{30}\right)$$
$$\chi_{1148}(227,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{47}{60}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{41}{120}\right)$$ $$e\left(\frac{39}{40}\right)$$ $$e\left(\frac{13}{40}\right)$$ $$e\left(\frac{11}{120}\right)$$ $$e\left(\frac{103}{120}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{17}{30}\right)$$
$$\chi_{1148}(299,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{1}{60}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{103}{120}\right)$$ $$e\left(\frac{17}{40}\right)$$ $$e\left(\frac{19}{40}\right)$$ $$e\left(\frac{13}{120}\right)$$ $$e\left(\frac{89}{120}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{1}{30}\right)$$
$$\chi_{1148}(311,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{59}{60}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{17}{120}\right)$$ $$e\left(\frac{23}{40}\right)$$ $$e\left(\frac{21}{40}\right)$$ $$e\left(\frac{107}{120}\right)$$ $$e\left(\frac{31}{120}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{29}{30}\right)$$
$$\chi_{1148}(339,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{29}{60}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{47}{120}\right)$$ $$e\left(\frac{33}{40}\right)$$ $$e\left(\frac{11}{40}\right)$$ $$e\left(\frac{77}{120}\right)$$ $$e\left(\frac{1}{120}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{29}{30}\right)$$
$$\chi_{1148}(395,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{11}{60}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{53}{120}\right)$$ $$e\left(\frac{27}{40}\right)$$ $$e\left(\frac{9}{40}\right)$$ $$e\left(\frac{23}{120}\right)$$ $$e\left(\frac{19}{120}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{11}{30}\right)$$
$$\chi_{1148}(423,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{53}{60}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{59}{120}\right)$$ $$e\left(\frac{21}{40}\right)$$ $$e\left(\frac{7}{40}\right)$$ $$e\left(\frac{89}{120}\right)$$ $$e\left(\frac{37}{120}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{23}{30}\right)$$
$$\chi_{1148}(439,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{1}{60}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{43}{120}\right)$$ $$e\left(\frac{37}{40}\right)$$ $$e\left(\frac{39}{40}\right)$$ $$e\left(\frac{73}{120}\right)$$ $$e\left(\frac{29}{120}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{1}{30}\right)$$
$$\chi_{1148}(479,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{53}{60}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{119}{120}\right)$$ $$e\left(\frac{1}{40}\right)$$ $$e\left(\frac{27}{40}\right)$$ $$e\left(\frac{29}{120}\right)$$ $$e\left(\frac{97}{120}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{23}{30}\right)$$
$$\chi_{1148}(507,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{11}{60}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{113}{120}\right)$$ $$e\left(\frac{7}{40}\right)$$ $$e\left(\frac{29}{40}\right)$$ $$e\left(\frac{83}{120}\right)$$ $$e\left(\frac{79}{120}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{11}{30}\right)$$
$$\chi_{1148}(563,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{29}{60}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{107}{120}\right)$$ $$e\left(\frac{13}{40}\right)$$ $$e\left(\frac{31}{40}\right)$$ $$e\left(\frac{17}{120}\right)$$ $$e\left(\frac{61}{120}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{29}{30}\right)$$
$$\chi_{1148}(591,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{59}{60}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{77}{120}\right)$$ $$e\left(\frac{3}{40}\right)$$ $$e\left(\frac{1}{40}\right)$$ $$e\left(\frac{47}{120}\right)$$ $$e\left(\frac{91}{120}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{29}{30}\right)$$
$$\chi_{1148}(663,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{37}{60}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{91}{120}\right)$$ $$e\left(\frac{29}{40}\right)$$ $$e\left(\frac{23}{40}\right)$$ $$e\left(\frac{1}{120}\right)$$ $$e\left(\frac{53}{120}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{7}{30}\right)$$
$$\chi_{1148}(675,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{47}{60}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{101}{120}\right)$$ $$e\left(\frac{19}{40}\right)$$ $$e\left(\frac{33}{40}\right)$$ $$e\left(\frac{71}{120}\right)$$ $$e\left(\frac{43}{120}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{17}{30}\right)$$
$$\chi_{1148}(691,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{43}{60}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{49}{120}\right)$$ $$e\left(\frac{31}{40}\right)$$ $$e\left(\frac{37}{40}\right)$$ $$e\left(\frac{19}{120}\right)$$ $$e\left(\frac{47}{120}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{13}{30}\right)$$
$$\chi_{1148}(703,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{23}{60}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{29}{120}\right)$$ $$e\left(\frac{11}{40}\right)$$ $$e\left(\frac{17}{40}\right)$$ $$e\left(\frac{119}{120}\right)$$ $$e\left(\frac{67}{120}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{23}{30}\right)$$
$$\chi_{1148}(719,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{7}{60}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{120}\right)$$ $$e\left(\frac{39}{40}\right)$$ $$e\left(\frac{13}{40}\right)$$ $$e\left(\frac{91}{120}\right)$$ $$e\left(\frac{23}{120}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{7}{30}\right)$$
$$\chi_{1148}(731,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{17}{60}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{71}{120}\right)$$ $$e\left(\frac{9}{40}\right)$$ $$e\left(\frac{3}{40}\right)$$ $$e\left(\frac{101}{120}\right)$$ $$e\left(\frac{73}{120}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{17}{30}\right)$$
$$\chi_{1148}(803,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{19}{60}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{97}{120}\right)$$ $$e\left(\frac{23}{40}\right)$$ $$e\left(\frac{21}{40}\right)$$ $$e\left(\frac{67}{120}\right)$$ $$e\left(\frac{71}{120}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{19}{30}\right)$$
$$\chi_{1148}(831,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{49}{60}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{7}{120}\right)$$ $$e\left(\frac{33}{40}\right)$$ $$e\left(\frac{11}{40}\right)$$ $$e\left(\frac{37}{120}\right)$$ $$e\left(\frac{41}{120}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{19}{30}\right)$$
$$\chi_{1148}(887,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{31}{60}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{13}{120}\right)$$ $$e\left(\frac{27}{40}\right)$$ $$e\left(\frac{9}{40}\right)$$ $$e\left(\frac{103}{120}\right)$$ $$e\left(\frac{59}{120}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{1}{30}\right)$$
$$\chi_{1148}(915,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{13}{60}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{19}{120}\right)$$ $$e\left(\frac{21}{40}\right)$$ $$e\left(\frac{7}{40}\right)$$ $$e\left(\frac{49}{120}\right)$$ $$e\left(\frac{77}{120}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{13}{30}\right)$$
$$\chi_{1148}(955,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{41}{60}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{23}{120}\right)$$ $$e\left(\frac{17}{40}\right)$$ $$e\left(\frac{19}{40}\right)$$ $$e\left(\frac{53}{120}\right)$$ $$e\left(\frac{49}{120}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{11}{30}\right)$$
$$\chi_{1148}(971,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{13}{60}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{79}{120}\right)$$ $$e\left(\frac{1}{40}\right)$$ $$e\left(\frac{27}{40}\right)$$ $$e\left(\frac{109}{120}\right)$$ $$e\left(\frac{17}{120}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{13}{30}\right)$$
$$\chi_{1148}(999,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{31}{60}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{73}{120}\right)$$ $$e\left(\frac{7}{40}\right)$$ $$e\left(\frac{29}{40}\right)$$ $$e\left(\frac{43}{120}\right)$$ $$e\left(\frac{119}{120}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{1}{30}\right)$$
$$\chi_{1148}(1055,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{49}{60}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{67}{120}\right)$$ $$e\left(\frac{13}{40}\right)$$ $$e\left(\frac{31}{40}\right)$$ $$e\left(\frac{97}{120}\right)$$ $$e\left(\frac{101}{120}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{19}{30}\right)$$
$$\chi_{1148}(1083,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{19}{60}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{37}{120}\right)$$ $$e\left(\frac{3}{40}\right)$$ $$e\left(\frac{1}{40}\right)$$ $$e\left(\frac{7}{120}\right)$$ $$e\left(\frac{11}{120}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{19}{30}\right)$$