# Properties

 Label 1859.170 Modulus $1859$ Conductor $11$ Order $5$ Real no Primitive no Minimal yes Parity even

# Learn more

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

sage: H = DirichletGroup(1859, base_ring=CyclotomicField(10))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([4,0]))

pari: [g,chi] = znchar(Mod(170,1859))

## Basic properties

 Modulus: $$1859$$ Conductor: $$11$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$5$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{11}(5,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 1859.h

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: $$\Q(\zeta_{5})$$ Fixed field: $$\Q(\zeta_{11})^+$$

## Values on generators

$$(508,1354)$$ → $$(e\left(\frac{2}{5}\right),1)$$

## Values

 $$a$$ $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$12$$ $$\chi_{ 1859 }(170, a)$$ $$1$$ $$1$$ $$e\left(\frac{2}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{2}{5}\right)$$ $$1$$ $$1$$
sage: chi.jacobi_sum(n)

$$\chi_{ 1859 }(170,a) \;$$ at $$\;a =$$ e.g. 2