# Properties

 Label 1287.ep Modulus $1287$ Conductor $1287$ Order $60$ Real no Primitive yes Minimal yes Parity odd

# Related objects

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

sage: H = DirichletGroup(1287, base_ring=CyclotomicField(60))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([20,48,25]))

sage: chi.galois_orbit()

pari: [g,chi] = znchar(Mod(58,1287))

pari: order = charorder(g,chi)

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

## Basic properties

 Modulus: $$1287$$ Conductor: $$1287$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$60$$ 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: $$\Q(\zeta_{60})$$ Fixed field: Number field defined by a degree 60 polynomial

## Characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$14$$ $$16$$ $$17$$ $$19$$
$$\chi_{1287}(58,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{37}{60}\right)$$ $$e\left(\frac{31}{60}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{29}{60}\right)$$
$$\chi_{1287}(115,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{11}{60}\right)$$ $$e\left(\frac{53}{60}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{7}{60}\right)$$
$$\chi_{1287}(202,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{43}{60}\right)$$ $$e\left(\frac{49}{60}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{11}{60}\right)$$
$$\chi_{1287}(223,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{17}{60}\right)$$ $$e\left(\frac{11}{60}\right)$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{49}{60}\right)$$
$$\chi_{1287}(466,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{23}{60}\right)$$ $$e\left(\frac{29}{60}\right)$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{31}{60}\right)$$
$$\chi_{1287}(526,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{49}{60}\right)$$ $$e\left(\frac{7}{60}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{53}{60}\right)$$
$$\chi_{1287}(553,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{7}{60}\right)$$ $$e\left(\frac{1}{60}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{1}{30}\right)$$ $$e\left(\frac{59}{60}\right)$$
$$\chi_{1287}(643,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{60}\right)$$ $$e\left(\frac{43}{60}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{17}{60}\right)$$
$$\chi_{1287}(691,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{7}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{29}{60}\right)$$ $$e\left(\frac{47}{60}\right)$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{13}{60}\right)$$
$$\chi_{1287}(808,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{41}{60}\right)$$ $$e\left(\frac{23}{60}\right)$$ $$e\left(\frac{9}{20}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{8}{15}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{23}{30}\right)$$ $$e\left(\frac{37}{60}\right)$$
$$\chi_{1287}(817,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{20}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{47}{60}\right)$$ $$e\left(\frac{41}{60}\right)$$ $$e\left(\frac{3}{20}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{11}{15}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{11}{30}\right)$$ $$e\left(\frac{19}{60}\right)$$
$$\chi_{1287}(994,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{13}{60}\right)$$ $$e\left(\frac{19}{60}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{19}{30}\right)$$ $$e\left(\frac{41}{60}\right)$$
$$\chi_{1287}(1021,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{19}{60}\right)$$ $$e\left(\frac{37}{60}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{7}{30}\right)$$ $$e\left(\frac{23}{60}\right)$$
$$\chi_{1287}(1138,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{13}{20}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{31}{60}\right)$$ $$e\left(\frac{13}{60}\right)$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{13}{15}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{47}{60}\right)$$
$$\chi_{1287}(1159,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{19}{20}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{53}{60}\right)$$ $$e\left(\frac{59}{60}\right)$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{14}{15}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{29}{30}\right)$$ $$e\left(\frac{1}{60}\right)$$
$$\chi_{1287}(1285,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{17}{20}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{59}{60}\right)$$ $$e\left(\frac{17}{60}\right)$$ $$e\left(\frac{11}{20}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{15}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{17}{30}\right)$$ $$e\left(\frac{43}{60}\right)$$