# Properties

 Label 287.be Modulus $287$ Conductor $287$ Order $120$ Real no Primitive yes Minimal yes Parity even

# Learn more

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

H = DirichletGroup(287, base_ring=CyclotomicField(120))

M = H._module

chi = DirichletCharacter(H, M([100,81]))

chi.galois_orbit()

[g,chi] = znchar(Mod(12,287))

order = charorder(g,chi)

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

## Basic properties

 Modulus: $$287$$ Conductor: $$287$$ 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: even 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 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$8$$ $$9$$ $$10$$ $$11$$ $$12$$
$$\chi_{287}(12,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{13}{60}\right)$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{1}{60}\right)$$ $$e\left(\frac{7}{40}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{43}{120}\right)$$ $$e\left(\frac{47}{120}\right)$$
$$\chi_{287}(17,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{47}{60}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{59}{60}\right)$$ $$e\left(\frac{13}{40}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{17}{120}\right)$$ $$e\left(\frac{13}{120}\right)$$
$$\chi_{287}(19,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{31}{60}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{7}{60}\right)$$ $$e\left(\frac{29}{40}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{1}{120}\right)$$ $$e\left(\frac{29}{120}\right)$$
$$\chi_{287}(24,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{47}{60}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{59}{60}\right)$$ $$e\left(\frac{33}{40}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{77}{120}\right)$$ $$e\left(\frac{73}{120}\right)$$
$$\chi_{287}(26,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{43}{60}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{31}{60}\right)$$ $$e\left(\frac{37}{40}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{73}{120}\right)$$ $$e\left(\frac{77}{120}\right)$$
$$\chi_{287}(47,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{19}{60}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{43}{60}\right)$$ $$e\left(\frac{21}{40}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{49}{120}\right)$$ $$e\left(\frac{101}{120}\right)$$
$$\chi_{287}(52,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{17}{60}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{29}{60}\right)$$ $$e\left(\frac{23}{40}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{107}{120}\right)$$ $$e\left(\frac{103}{120}\right)$$
$$\chi_{287}(54,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{49}{60}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{13}{60}\right)$$ $$e\left(\frac{11}{40}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{79}{120}\right)$$ $$e\left(\frac{11}{120}\right)$$
$$\chi_{287}(75,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{60}\right)$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{37}{60}\right)$$ $$e\left(\frac{39}{40}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{91}{120}\right)$$ $$e\left(\frac{119}{120}\right)$$
$$\chi_{287}(89,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{60}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{37}{60}\right)$$ $$e\left(\frac{19}{40}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{31}{120}\right)$$ $$e\left(\frac{59}{120}\right)$$
$$\chi_{287}(94,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{53}{60}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{41}{60}\right)$$ $$e\left(\frac{7}{40}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{83}{120}\right)$$ $$e\left(\frac{7}{120}\right)$$
$$\chi_{287}(101,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{11}{60}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{47}{60}\right)$$ $$e\left(\frac{29}{40}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{41}{120}\right)$$ $$e\left(\frac{109}{120}\right)$$
$$\chi_{287}(108,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{23}{60}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{11}{60}\right)$$ $$e\left(\frac{37}{40}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{113}{120}\right)$$ $$e\left(\frac{37}{120}\right)$$
$$\chi_{287}(110,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{49}{60}\right)$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{13}{60}\right)$$ $$e\left(\frac{31}{40}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{19}{120}\right)$$ $$e\left(\frac{71}{120}\right)$$
$$\chi_{287}(117,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{19}{60}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{43}{60}\right)$$ $$e\left(\frac{1}{40}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{109}{120}\right)$$ $$e\left(\frac{41}{120}\right)$$
$$\chi_{287}(129,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{59}{60}\right)$$ $$e\left(\frac{13}{24}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{23}{60}\right)$$ $$e\left(\frac{21}{40}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{89}{120}\right)$$ $$e\left(\frac{61}{120}\right)$$
$$\chi_{287}(136,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{29}{60}\right)$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{53}{60}\right)$$ $$e\left(\frac{11}{40}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{119}{120}\right)$$ $$e\left(\frac{91}{120}\right)$$
$$\chi_{287}(138,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{43}{60}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{31}{60}\right)$$ $$e\left(\frac{17}{40}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{13}{120}\right)$$ $$e\left(\frac{17}{120}\right)$$
$$\chi_{287}(145,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{31}{60}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{7}{60}\right)$$ $$e\left(\frac{9}{40}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{61}{120}\right)$$ $$e\left(\frac{89}{120}\right)$$
$$\chi_{287}(152,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{13}{60}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{1}{60}\right)$$ $$e\left(\frac{27}{40}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{103}{120}\right)$$ $$e\left(\frac{107}{120}\right)$$
$$\chi_{287}(157,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{41}{60}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{17}{60}\right)$$ $$e\left(\frac{39}{40}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{11}{120}\right)$$ $$e\left(\frac{79}{120}\right)$$
$$\chi_{287}(171,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{41}{60}\right)$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{17}{60}\right)$$ $$e\left(\frac{19}{40}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{71}{120}\right)$$ $$e\left(\frac{19}{120}\right)$$
$$\chi_{287}(192,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{29}{60}\right)$$ $$e\left(\frac{7}{24}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{53}{60}\right)$$ $$e\left(\frac{31}{40}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{59}{120}\right)$$ $$e\left(\frac{31}{120}\right)$$
$$\chi_{287}(194,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{37}{60}\right)$$ $$e\left(\frac{11}{24}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{49}{60}\right)$$ $$e\left(\frac{3}{40}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{7}{120}\right)$$ $$e\left(\frac{83}{120}\right)$$
$$\chi_{287}(199,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{59}{60}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{23}{60}\right)$$ $$e\left(\frac{1}{40}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{29}{120}\right)$$ $$e\left(\frac{1}{120}\right)$$
$$\chi_{287}(220,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{23}{60}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{11}{60}\right)$$ $$e\left(\frac{17}{40}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{53}{120}\right)$$ $$e\left(\frac{97}{120}\right)$$
$$\chi_{287}(222,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{7}{60}\right)$$ $$e\left(\frac{5}{24}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{19}{60}\right)$$ $$e\left(\frac{13}{40}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{97}{120}\right)$$ $$e\left(\frac{53}{120}\right)$$
$$\chi_{287}(227,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{11}{60}\right)$$ $$e\left(\frac{1}{24}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{47}{60}\right)$$ $$e\left(\frac{9}{40}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{101}{120}\right)$$ $$e\left(\frac{49}{120}\right)$$
$$\chi_{287}(229,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{7}{60}\right)$$ $$e\left(\frac{17}{24}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{19}{60}\right)$$ $$e\left(\frac{33}{40}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{37}{120}\right)$$ $$e\left(\frac{113}{120}\right)$$
$$\chi_{287}(234,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{53}{60}\right)$$ $$e\left(\frac{19}{24}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{41}{60}\right)$$ $$e\left(\frac{27}{40}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{23}{120}\right)$$ $$e\left(\frac{67}{120}\right)$$
$$\chi_{287}(257,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{37}{60}\right)$$ $$e\left(\frac{23}{24}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{49}{60}\right)$$ $$e\left(\frac{23}{40}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{67}{120}\right)$$ $$e\left(\frac{23}{120}\right)$$