# Properties

 Label 165.r Modulus $165$ Conductor $165$ Order $10$ Real no Primitive yes Minimal yes Parity even

# Related objects

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

H = DirichletGroup(165, base_ring=CyclotomicField(10))

M = H._module

chi = DirichletCharacter(H, M([5,5,7]))

chi.galois_orbit()

[g,chi] = znchar(Mod(29,165))

order = charorder(g,chi)

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

## Basic properties

 Modulus: $$165$$ Conductor: $$165$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$10$$ 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: $$\Q(\zeta_{5})$$ Fixed field: 10.10.1790566527853125.1

## Characters in Galois orbit

Character $$-1$$ $$1$$ $$2$$ $$4$$ $$7$$ $$8$$ $$13$$ $$14$$ $$16$$ $$17$$ $$19$$ $$23$$
$$\chi_{165}(29,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$1$$
$$\chi_{165}(74,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$1$$
$$\chi_{165}(134,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$1$$
$$\chi_{165}(149,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{1}{10}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$1$$