# Properties

 Label 1792.209 Modulus $1792$ Conductor $448$ Order $16$ Real no Primitive no Minimal no Parity odd

# Related objects

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

sage: H = DirichletGroup(1792, base_ring=CyclotomicField(16))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,11,8]))

pari: [g,chi] = znchar(Mod(209,1792))

## Basic properties

 Modulus: $$1792$$ Conductor: $$448$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$16$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{448}(349,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: no Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 1792.bf

sage: chi.galois_orbit()

pari: order = charorder(g,chi)

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

## Related number fields

 Field of values: $$\Q(\zeta_{16})$$ Fixed field: 16.0.3484608386920116940487669055488.4

## Values on generators

$$(1023,1541,1025)$$ → $$(1,e\left(\frac{11}{16}\right),-1)$$

## Values

 $$a$$ $$-1$$ $$1$$ $$3$$ $$5$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$19$$ $$23$$ $$25$$ $$\chi_{ 1792 }(209, a)$$ $$-1$$ $$1$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{3}{16}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{7}{16}\right)$$ $$e\left(\frac{13}{16}\right)$$ $$-i$$ $$-i$$ $$e\left(\frac{5}{16}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 1792 }(209,a) \;$$ at $$\;a =$$ e.g. 2