# Properties

 Label 441.bg Modulus $441$ Conductor $147$ Order $42$ Real no Primitive no Minimal yes Parity even

# Related objects

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

H = DirichletGroup(441, base_ring=CyclotomicField(42))

M = H._module

chi = DirichletCharacter(H, M([21,25]))

chi.galois_orbit()

[g,chi] = znchar(Mod(17,441))

order = charorder(g,chi)

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

## Basic properties

 Modulus: $$441$$ Conductor: $$147$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$42$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from 147.o 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: $$\Q(\zeta_{21})$$ Fixed field: $$\Q(\zeta_{147})^+$$

## Characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$8$$ $$10$$ $$11$$ $$13$$ $$16$$ $$17$$ $$19$$
$$\chi_{441}(17,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{441}(26,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{441}(89,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{441}(143,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{441}(152,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{441}(206,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{1}{42}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{441}(269,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{31}{42}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{441}(278,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{29}{42}\right)$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{441}(332,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{441}(341,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{25}{42}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{23}{42}\right)$$ $$e\left(\frac{11}{42}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{1}{6}\right)$$
$$\chi_{441}(395,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{5}{42}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{19}{42}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{5}{6}\right)$$
$$\chi_{441}(404,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{13}{42}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{17}{42}\right)$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{1}{6}\right)$$