# Properties

 Label 171.z Modulus $171$ Conductor $171$ Order $18$ Real no Primitive yes Minimal yes Parity odd

# Related objects

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

H = DirichletGroup(171, base_ring=CyclotomicField(18))

M = H._module

chi = DirichletCharacter(H, M([15,16]))

chi.galois_orbit()

[g,chi] = znchar(Mod(5,171))

order = charorder(g,chi)

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

## Basic properties

 Modulus: $$171$$ Conductor: $$171$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$18$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: yes 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: $$\Q(\zeta_{9})$$ Fixed field: Number field defined by a degree 18 polynomial

## Characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$11$$ $$13$$ $$14$$ $$16$$
$$\chi_{171}(5,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$-1$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$
$$\chi_{171}(47,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$-1$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{4}{9}\right)$$
$$\chi_{171}(101,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$-1$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$
$$\chi_{171}(131,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{7}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$-1$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{5}{9}\right)$$
$$\chi_{171}(137,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{5}{18}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$-1$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$
$$\chi_{171}(149,\cdot)$$ $$-1$$ $$1$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$-1$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{13}{18}\right)$$ $$e\left(\frac{2}{9}\right)$$