# Properties

 Label 405.r Modulus $405$ Conductor $135$ Order $36$ Real no Primitive no Minimal no Parity even

# Related objects

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

H = DirichletGroup(405, base_ring=CyclotomicField(36))

M = H._module

chi = DirichletCharacter(H, M([2,27]))

chi.galois_orbit()

[g,chi] = znchar(Mod(8,405))

order = charorder(g,chi)

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

## Basic properties

 Modulus: $$405$$ Conductor: $$135$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$36$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from 135.q sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: no Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Related number fields

 Field of values: $$\Q(\zeta_{36})$$ Fixed field: $$\Q(\zeta_{135})^+$$

## Characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$4$$ $$7$$ $$8$$ $$11$$ $$13$$ $$14$$ $$16$$ $$17$$ $$19$$
$$\chi_{405}(8,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{29}{36}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{23}{36}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{25}{36}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{405}(17,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{23}{36}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{405}(62,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{23}{36}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{17}{36}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{405}(98,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{13}{36}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{19}{36}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{5}{36}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{405}(143,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{5}{36}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{13}{36}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{405}(152,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{7}{36}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{13}{36}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{11}{36}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{405}(197,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{29}{36}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{19}{36}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{405}(233,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{25}{36}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{31}{36}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{29}{36}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{405}(278,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{17}{36}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{11}{36}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{405}(287,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{19}{36}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{25}{36}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{405}(332,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{11}{36}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{5}{36}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{7}{36}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{405}(368,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{36}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{7}{36}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{17}{36}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$