# Properties

 Label 441.z Modulus $441$ Conductor $441$ Order $21$ Real no Primitive yes 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([14,10]))

chi.galois_orbit()

[g,chi] = znchar(Mod(4,441))

order = charorder(g,chi)

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

## Basic properties

 Modulus: $$441$$ Conductor: $$441$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$21$$ 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: $$\Q(\zeta_{21})$$ Fixed field: Number field defined by a degree 21 polynomial

## Characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$8$$ $$10$$ $$11$$ $$13$$ $$16$$ $$17$$ $$19$$
$$\chi_{441}(4,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{441}(16,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{441}(130,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{441}(142,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{441}(193,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{441}(205,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{441}(256,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{2}{21}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{17}{21}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{441}(268,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{4}{21}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{13}{21}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{441}(319,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{441}(331,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{20}{21}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{19}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{441}(382,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{8}{21}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{441}(394,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{11}{21}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{16}{21}\right)$$ $$e\left(\frac{1}{21}\right)$$ $$e\left(\frac{10}{21}\right)$$ $$e\left(\frac{2}{3}\right)$$