# Properties

 Label 819.739 Modulus $819$ Conductor $91$ Order $12$ Real no Primitive no Minimal yes Parity odd

# Related objects

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

sage: H = DirichletGroup(819, base_ring=CyclotomicField(12))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,8,7]))

pari: [g,chi] = znchar(Mod(739,819))

## Basic properties

 Modulus: $$819$$ Conductor: $$91$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$12$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{91}(11,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 819.ff

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: $$\Q(\zeta_{12})$$ Fixed field: 12.0.10331448031704891637.2

## Values on generators

$$(92,703,379)$$ → $$(1,e\left(\frac{2}{3}\right),e\left(\frac{7}{12}\right))$$

## Values

 $$a$$ $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$8$$ $$10$$ $$11$$ $$16$$ $$17$$ $$19$$ $$20$$ $$\chi_{ 819 }(739, a)$$ $$-1$$ $$1$$ $$e\left(\frac{11}{12}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{7}{12}\right)$$ $$-i$$ $$-1$$ $$-i$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$i$$ $$e\left(\frac{5}{12}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 819 }(739,a) \;$$ at $$\;a =$$ e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 819 }(739,·) )\;$$ at $$\;a =$$ e.g. 2

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 819 }(739,·),\chi_{ 819 }(n,·)) \;$$ for $$\; n =$$ e.g. 1

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 819 }(739,·)) \;$$ at $$\; a,b =$$ e.g. 1,2