# Properties

 Label 162.g Modulus $162$ Conductor $81$ Order $27$ Real no Primitive no Minimal yes Parity even

# Related objects

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

H = DirichletGroup(162, base_ring=CyclotomicField(54))

M = H._module

chi = DirichletCharacter(H, M([16]))

chi.galois_orbit()

[g,chi] = znchar(Mod(7,162))

order = charorder(g,chi)

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

## Basic properties

 Modulus: $$162$$ Conductor: $$81$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$27$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from 81.g 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 $$-1$$ $$1$$ $$5$$ $$7$$ $$11$$ $$13$$ $$17$$ $$19$$ $$23$$ $$25$$ $$29$$ $$31$$
$$\chi_{162}(7,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{20}{27}\right)$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{7}{27}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{25}{27}\right)$$
$$\chi_{162}(13,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{11}{27}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{13}{27}\right)$$ $$e\left(\frac{26}{27}\right)$$
$$\chi_{162}(25,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{1}{27}\right)$$
$$\chi_{162}(31,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{1}{27}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{11}{27}\right)$$
$$\chi_{162}(43,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{8}{27}\right)$$ $$e\left(\frac{7}{27}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{13}{27}\right)$$ $$e\left(\frac{20}{27}\right)$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{4}{27}\right)$$
$$\chi_{162}(49,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{13}{27}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{20}{27}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{7}{27}\right)$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{23}{27}\right)$$
$$\chi_{162}(61,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{4}{27}\right)$$ $$e\left(\frac{11}{27}\right)$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{8}{27}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{7}{27}\right)$$
$$\chi_{162}(67,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{20}{27}\right)$$ $$e\left(\frac{1}{27}\right)$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{13}{27}\right)$$ $$e\left(\frac{4}{27}\right)$$ $$e\left(\frac{8}{27}\right)$$
$$\chi_{162}(79,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{8}{27}\right)$$ $$e\left(\frac{20}{27}\right)$$ $$e\left(\frac{4}{27}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{10}{27}\right)$$
$$\chi_{162}(85,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{13}{27}\right)$$ $$e\left(\frac{8}{27}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{11}{27}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{20}{27}\right)$$
$$\chi_{162}(97,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{11}{27}\right)$$ $$e\left(\frac{20}{27}\right)$$ $$e\left(\frac{13}{27}\right)$$
$$\chi_{162}(103,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{4}{27}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{5}{27}\right)$$
$$\chi_{162}(115,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{13}{27}\right)$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{1}{27}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{8}{27}\right)$$ $$e\left(\frac{16}{27}\right)$$
$$\chi_{162}(121,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{19}{27}\right)$$ $$e\left(\frac{7}{27}\right)$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{8}{27}\right)$$ $$e\left(\frac{4}{27}\right)$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{17}{27}\right)$$
$$\chi_{162}(133,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{7}{27}\right)$$ $$e\left(\frac{26}{27}\right)$$ $$e\left(\frac{11}{27}\right)$$ $$e\left(\frac{13}{27}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{27}\right)$$ $$e\left(\frac{14}{27}\right)$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{19}{27}\right)$$
$$\chi_{162}(139,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{7}{27}\right)$$ $$e\left(\frac{4}{27}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{20}{27}\right)$$ $$e\left(\frac{10}{27}\right)$$ $$e\left(\frac{1}{27}\right)$$ $$e\left(\frac{2}{27}\right)$$
$$\chi_{162}(151,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{27}\right)$$ $$e\left(\frac{23}{27}\right)$$ $$e\left(\frac{17}{27}\right)$$ $$e\left(\frac{25}{27}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{4}{27}\right)$$ $$e\left(\frac{2}{27}\right)$$ $$e\left(\frac{11}{27}\right)$$ $$e\left(\frac{22}{27}\right)$$
$$\chi_{162}(157,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{8}{27}\right)$$ $$e\left(\frac{22}{27}\right)$$ $$e\left(\frac{1}{27}\right)$$ $$e\left(\frac{11}{27}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{5}{27}\right)$$ $$e\left(\frac{16}{27}\right)$$ $$e\left(\frac{7}{27}\right)$$ $$e\left(\frac{14}{27}\right)$$