# Properties

 Label 837.673 Modulus $837$ Conductor $837$ Order $90$ Real no Primitive yes Minimal yes Parity odd

# Related objects

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

H = DirichletGroup(837, base_ring=CyclotomicField(90))

M = H._module

chi = DirichletCharacter(H, M([50,51]))

pari: [g,chi] = znchar(Mod(673,837))

## Basic properties

 Modulus: $$837$$ Conductor: $$837$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$90$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: yes sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 837.cl

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

$$(218,406)$$ → $$(e\left(\frac{5}{9}\right),e\left(\frac{17}{30}\right))$$

## First values

 $$a$$ $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$11$$ $$13$$ $$14$$ $$16$$ $$\chi_{ 837 }(673, a)$$ $$-1$$ $$1$$ $$e\left(\frac{7}{45}\right)$$ $$e\left(\frac{14}{45}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{34}{45}\right)$$ $$e\left(\frac{7}{15}\right)$$ $$e\left(\frac{4}{15}\right)$$ $$e\left(\frac{23}{90}\right)$$ $$e\left(\frac{61}{90}\right)$$ $$e\left(\frac{41}{45}\right)$$ $$e\left(\frac{28}{45}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 837 }(673,a) \;$$ at $$\;a =$$ e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 837 }(673,·) )\;$$ at $$\;a =$$ e.g. 2

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 837 }(673,·),\chi_{ 837 }(n,·)) \;$$ for $$\; n =$$ e.g. 1

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 837 }(673,·)) \;$$ at $$\; a,b =$$ e.g. 1,2