# Properties

 Label 1805.49 Modulus $1805$ Conductor $1805$ Order $114$ Real no Primitive yes Minimal yes Parity even

# Related objects

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

sage: H = DirichletGroup(1805, base_ring=CyclotomicField(114))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([57,100]))

pari: [g,chi] = znchar(Mod(49,1805))

## Basic properties

 Modulus: $$1805$$ Conductor: $$1805$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$114$$ 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 1805.ba

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: $\Q(\zeta_{57})$ Fixed field: Number field defined by a degree 114 polynomial (not computed)

## Values on generators

$$(362,1446)$$ → $$(-1,e\left(\frac{50}{57}\right))$$

## Values

 $$a$$ $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$6$$ $$7$$ $$8$$ $$9$$ $$11$$ $$12$$ $$13$$ $$\chi_{ 1805 }(49, a)$$ $$1$$ $$1$$ $$e\left(\frac{43}{114}\right)$$ $$e\left(\frac{49}{114}\right)$$ $$e\left(\frac{43}{57}\right)$$ $$e\left(\frac{46}{57}\right)$$ $$e\left(\frac{3}{38}\right)$$ $$e\left(\frac{5}{38}\right)$$ $$e\left(\frac{49}{57}\right)$$ $$e\left(\frac{9}{19}\right)$$ $$e\left(\frac{7}{38}\right)$$ $$e\left(\frac{11}{114}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 1805 }(49,a) \;$$ at $$\;a =$$ e.g. 2