# Properties

 Label 1045.197 Modulus $1045$ Conductor $1045$ Order $12$ Real no Primitive yes Minimal yes Parity even

# Related objects

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

H = DirichletGroup(1045, base_ring=CyclotomicField(12))

M = H._module

chi = DirichletCharacter(H, M([3,6,4]))

pari: [g,chi] = znchar(Mod(197,1045))

## Basic properties

 Modulus: $$1045$$ Conductor: $$1045$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$12$$ 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)

## Galois orbit 1045.bg

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: $$\Q(\zeta_{12})$$ Fixed field: 12.12.58764488133744142578125.1

## Values on generators

$$(837,761,496)$$ → $$(i,-1,e\left(\frac{1}{3}\right))$$

## First values

 $$a$$ $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$6$$ $$7$$ $$8$$ $$9$$ $$12$$ $$13$$ $$14$$ $$\chi_{ 1045 }(197, a)$$ $$1$$ $$1$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$-i$$ $$i$$ $$e\left(\frac{1}{6}\right)$$ $$i$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 1045 }(197,a) \;$$ at $$\;a =$$ e.g. 2