# Properties

 Label 1045.112 Modulus $1045$ Conductor $1045$ Order $180$ Real no Primitive yes Minimal yes Parity even

# Learn more

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

H = DirichletGroup(1045, base_ring=CyclotomicField(180))

M = H._module

chi = DirichletCharacter(H, M([45,18,100]))

pari: [g,chi] = znchar(Mod(112,1045))

## Basic properties

 Modulus: $$1045$$ Conductor: $$1045$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$180$$ 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.cs

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: $\Q(\zeta_{180})$ Fixed field: Number field defined by a degree 180 polynomial (not computed)

## Values on generators

$$(837,761,496)$$ → $$(i,e\left(\frac{1}{10}\right),e\left(\frac{5}{9}\right))$$

## First values

 $$a$$ $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$6$$ $$7$$ $$8$$ $$9$$ $$12$$ $$13$$ $$14$$ $$\chi_{ 1045 }(112, a)$$ $$1$$ $$1$$ $$e\left(\frac{163}{180}\right)$$ $$e\left(\frac{139}{180}\right)$$ $$e\left(\frac{73}{90}\right)$$ $$e\left(\frac{61}{90}\right)$$ $$e\left(\frac{17}{60}\right)$$ $$e\left(\frac{43}{60}\right)$$ $$e\left(\frac{49}{90}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{113}{180}\right)$$ $$e\left(\frac{17}{90}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 1045 }(112,a) \;$$ at $$\;a =$$ e.g. 2