# Properties

 Label 1638.1171 Modulus $1638$ Conductor $7$ Order $3$ Real no Primitive no Minimal yes Parity even

# Related objects

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

H = DirichletGroup(1638, base_ring=CyclotomicField(6))

M = H._module

chi = DirichletCharacter(H, M([0,2,0]))

pari: [g,chi] = znchar(Mod(1171,1638))

## Basic properties

 Modulus: $$1638$$ Conductor: $$7$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$3$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{7}(2,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 1638.j

sage: chi.galois_orbit()

order = charorder(g,chi)

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

## Related number fields

 Field of values: $$\mathbb{Q}(\zeta_3)$$ Fixed field: $$\Q(\zeta_{7})^+$$

## Values on generators

$$(911,703,379)$$ → $$(1,e\left(\frac{1}{3}\right),1)$$

## First values

 $$a$$ $$-1$$ $$1$$ $$5$$ $$11$$ $$17$$ $$19$$ $$23$$ $$25$$ $$29$$ $$31$$ $$37$$ $$41$$ $$\chi_{ 1638 }(1171, a)$$ $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$
sage: chi.jacobi_sum(n)

$$\chi_{ 1638 }(1171,a) \;$$ at $$\;a =$$ e.g. 2