# Properties

 Label 847.244 Modulus $847$ Conductor $847$ Order $110$ Real no Primitive yes Minimal yes Parity even

# Related objects

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

H = DirichletGroup(847, base_ring=CyclotomicField(110))

M = H._module

chi = DirichletCharacter(H, M([55,1]))

pari: [g,chi] = znchar(Mod(244,847))

## Basic properties

 Modulus: $$847$$ Conductor: $$847$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$110$$ 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 847.ba

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: $\Q(\zeta_{55})$ Fixed field: Number field defined by a degree 110 polynomial (not computed)

## Values on generators

$$(122,365)$$ → $$(-1,e\left(\frac{1}{110}\right))$$

## First values

 $$a$$ $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$8$$ $$9$$ $$10$$ $$12$$ $$13$$ $$\chi_{ 847 }(244, a)$$ $$1$$ $$1$$ $$e\left(\frac{1}{110}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{1}{55}\right)$$ $$e\left(\frac{19}{110}\right)$$ $$e\left(\frac{17}{55}\right)$$ $$e\left(\frac{3}{110}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{2}{11}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{23}{55}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 847 }(244,a) \;$$ at $$\;a =$$ e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 847 }(244,·) )\;$$ at $$\;a =$$ e.g. 2

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 847 }(244,·),\chi_{ 847 }(n,·)) \;$$ for $$\; n =$$ e.g. 1

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 847 }(244,·)) \;$$ at $$\; a,b =$$ e.g. 1,2