# Properties

 Label 1045.78 Modulus $1045$ Conductor $95$ Order $36$ Real no Primitive no Minimal yes Parity even

# Learn more

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

H = DirichletGroup(1045, base_ring=CyclotomicField(36))

M = H._module

chi = DirichletCharacter(H, M([27,0,2]))

pari: [g,chi] = znchar(Mod(78,1045))

## Basic properties

 Modulus: $$1045$$ Conductor: $$95$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$36$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{95}(78,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 1045.cc

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: $$\Q(\zeta_{36})$$ Fixed field: $$\Q(\zeta_{95})^+$$

## Values on generators

$$(837,761,496)$$ → $$(-i,1,e\left(\frac{1}{18}\right))$$

## First values

 $$a$$ $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$6$$ $$7$$ $$8$$ $$9$$ $$12$$ $$13$$ $$14$$ $$\chi_{ 1045 }(78, a)$$ $$1$$ $$1$$ $$e\left(\frac{29}{36}\right)$$ $$e\left(\frac{35}{36}\right)$$ $$e\left(\frac{11}{18}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$e\left(\frac{19}{36}\right)$$ $$e\left(\frac{8}{9}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 1045 }(78,a) \;$$ at $$\;a =$$ e.g. 2