# Properties

 Label 1045.41 Modulus $1045$ Conductor $209$ Order $90$ Real no Primitive no Minimal yes Parity even

# Related objects

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

H = DirichletGroup(1045, base_ring=CyclotomicField(90))

M = H._module

chi = DirichletCharacter(H, M([0,27,65]))

pari: [g,chi] = znchar(Mod(41,1045))

## Basic properties

 Modulus: $$1045$$ Conductor: $$209$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$90$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{209}(41,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 1045.cm

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: $\Q(\zeta_{45})$ Fixed field: Number field defined by a degree 90 polynomial

## Values on generators

$$(837,761,496)$$ → $$(1,e\left(\frac{3}{10}\right),e\left(\frac{13}{18}\right))$$

## Values

 $$a$$ $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$6$$ $$7$$ $$8$$ $$9$$ $$12$$ $$13$$ $$14$$ $$\chi_{ 1045 }(41, a)$$ $$1$$ $$1$$ $$e\left(\frac{1}{45}\right)$$ $$e\left(\frac{71}{90}\right)$$ $$e\left(\frac{2}{45}\right)$$ $$e\left(\frac{73}{90}\right)$$ $$e\left(\frac{13}{30}\right)$$ $$e\left(\frac{1}{15}\right)$$ $$e\left(\frac{26}{45}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{41}{45}\right)$$ $$e\left(\frac{41}{90}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 1045 }(41,a) \;$$ at $$\;a =$$ e.g. 2