# Properties

 Label 441.v Modulus $441$ Conductor $49$ Order $14$ Real no Primitive no Minimal yes Parity odd

# Related objects

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

H = DirichletGroup(441, base_ring=CyclotomicField(14))

M = H._module

chi = DirichletCharacter(H, M([0,9]))

chi.galois_orbit()

[g,chi] = znchar(Mod(55,441))

order = charorder(g,chi)

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

## Basic properties

 Modulus: $$441$$ Conductor: $$49$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$14$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from 49.f sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Related number fields

 Field of values: $$\Q(\zeta_{7})$$ Fixed field: 14.0.1341068619663964900807.1

## Characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$8$$ $$10$$ $$11$$ $$13$$ $$16$$ $$17$$ $$19$$
$$\chi_{441}(55,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$-1$$
$$\chi_{441}(118,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$-1$$
$$\chi_{441}(181,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$-1$$
$$\chi_{441}(307,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{3}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$-1$$
$$\chi_{441}(370,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{5}{7}\right)$$ $$e\left(\frac{1}{14}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$-1$$
$$\chi_{441}(433,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{4}{7}\right)$$ $$e\left(\frac{5}{14}\right)$$ $$e\left(\frac{6}{7}\right)$$ $$e\left(\frac{9}{14}\right)$$ $$e\left(\frac{2}{7}\right)$$ $$e\left(\frac{11}{14}\right)$$ $$e\left(\frac{1}{7}\right)$$ $$e\left(\frac{13}{14}\right)$$ $$-1$$