Properties

 Label 81.g Modulus $81$ Conductor $81$ Order $27$ Real no Primitive yes Minimal yes Parity even

Related objects

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

H = DirichletGroup(81, base_ring=CyclotomicField(54))

M = H._module

chi = DirichletCharacter(H, M([2]))

chi.galois_orbit()

[g,chi] = znchar(Mod(4,81))

order = charorder(g,chi)

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

Basic properties

 Modulus: $$81$$ Conductor: $$81$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$27$$ 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: $$\Q(\zeta_{27})$$ Fixed field: Number field defined by a degree 27 polynomial

Characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$11$$ $$13$$ $$14$$ $$16$$
$$\chi_{81}(4,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{27}\right)$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{13}{27}\right)$$ $$e\left(\frac{8}{27}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{4}{27}\right)$$
$$\chi_{81}(7,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{8}{27}\right)$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{20}{27}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{1}{27}\right)$$ $$e\left(\frac{5}{27}\right)$$
$$\chi_{81}(13,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{4}{27}\right)$$ $$e\left(\frac{8}{27}\right)$$ $$e\left(\frac{11}{27}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{16}{27}\right)$$
$$\chi_{81}(16,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{4}{27}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{7}{27}\right)$$ $$e\left(\frac{8}{27}\right)$$
$$\chi_{81}(22,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{7}{27}\right)$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{4}{27}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{11}{27}\right)$$ $$e\left(\frac{1}{27}\right)$$
$$\chi_{81}(25,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{13}{27}\right)$$ $$e\left(\frac{11}{27}\right)$$
$$\chi_{81}(31,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{20}{27}\right)$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{8}{27}\right)$$ $$e\left(\frac{13}{27}\right)$$
$$\chi_{81}(34,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{7}{27}\right)$$ $$e\left(\frac{13}{27}\right)$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{1}{27}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{14}{27}\right)$$
$$\chi_{81}(40,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{13}{27}\right)$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{7}{27}\right)$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{25}{27}\right)$$
$$\chi_{81}(43,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{11}{27}\right)$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{8}{27}\right)$$ $$e\left(\frac{7}{27}\right)$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{17}{27}\right)$$
$$\chi_{81}(49,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{13}{27}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{20}{27}\right)$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{10}{27}\right)$$
$$\chi_{81}(52,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{7}{27}\right)$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{11}{27}\right)$$ $$e\left(\frac{13}{27}\right)$$ $$e\left(\frac{4}{27}\right)$$ $$e\left(\frac{20}{27}\right)$$
$$\chi_{81}(58,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{11}{27}\right)$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{7}{27}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{4}{27}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{22}{27}\right)$$
$$\chi_{81}(61,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{4}{27}\right)$$ $$e\left(\frac{11}{27}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{23}{27}\right)$$
$$\chi_{81}(67,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{20}{27}\right)$$ $$e\left(\frac{1}{27}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{7}{27}\right)$$
$$\chi_{81}(70,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{20}{27}\right)$$ $$e\left(\frac{13}{27}\right)$$ $$e\left(\frac{1}{27}\right)$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{26}{27}\right)$$
$$\chi_{81}(76,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{8}{27}\right)$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{27}\right)$$ $$e\left(\frac{11}{27}\right)$$ $$e\left(\frac{20}{27}\right)$$ $$e\left(\frac{19}{27}\right)$$
$$\chi_{81}(79,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{1}{27}\right)$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{8}{27}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{20}{27}\right)$$ $$e\left(\frac{4}{27}\right)$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{2}{27}\right)$$