# Properties

 Label 76.i Modulus $76$ Conductor $19$ Order $9$ Real no Primitive no Minimal yes Parity even

# Related objects

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

H = DirichletGroup(76, base_ring=CyclotomicField(18))

M = H._module

chi = DirichletCharacter(H, M([0,16]))

chi.galois_orbit()

[g,chi] = znchar(Mod(5,76))

order = charorder(g,chi)

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

## Basic properties

 Modulus: $$76$$ Conductor: $$19$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$9$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from 19.e 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: $$\Q(\zeta_{9})$$ Fixed field: $$\Q(\zeta_{19})^+$$

## Characters in Galois orbit

Character $$-1$$ $$1$$ $$3$$ $$5$$ $$7$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$21$$ $$23$$
$$\chi_{76}(5,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$
$$\chi_{76}(9,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$
$$\chi_{76}(17,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$
$$\chi_{76}(25,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$
$$\chi_{76}(61,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$
$$\chi_{76}(73,\cdot)$$ $$1$$ $$1$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{5}{9}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{2}{9}\right)$$ $$e\left(\frac{4}{9}\right)$$