# Properties

 Label 111.m Modulus $111$ Conductor $111$ Order $12$ Real no Primitive yes Minimal yes Parity even

# Related objects

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

H = DirichletGroup(111, base_ring=CyclotomicField(12))

M = H._module

chi = DirichletCharacter(H, M([6,1]))

chi.galois_orbit()

[g,chi] = znchar(Mod(8,111))

order = charorder(g,chi)

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

## Basic properties

 Modulus: $$111$$ Conductor: $$111$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$12$$ 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: $$\Q(\zeta_{12})$$ Fixed field: 12.12.129701946277226641077.1

## Characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$11$$ $$13$$ $$14$$ $$16$$
$$\chi_{111}(8,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$-i$$ $$1$$ $$1$$ $$e\left(\frac{11}{12}\right)$$ $$i$$ $$e\left(\frac{1}{3}\right)$$
$$\chi_{111}(14,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$i$$ $$1$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$-i$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{111}(23,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$-i$$ $$1$$ $$1$$ $$e\left(\frac{7}{12}\right)$$ $$i$$ $$e\left(\frac{2}{3}\right)$$
$$\chi_{111}(29,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$i$$ $$1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$-i$$ $$e\left(\frac{1}{3}\right)$$