# Properties

 Label 184.51 Modulus $184$ Conductor $184$ Order $22$ Real no Primitive yes Minimal yes Parity even

# Learn more

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

sage: H = DirichletGroup(184, base_ring=CyclotomicField(22))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([11,11,1]))

pari: [g,chi] = znchar(Mod(51,184))

## Basic properties

 Modulus: $$184$$ Conductor: $$184$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$22$$ 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 184.j

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: $$\Q(\zeta_{11})$$ Fixed field: 22.22.339058325839400057321133061640411938816.1

## Values on generators

$$(47,93,97)$$ → $$(-1,-1,e\left(\frac{1}{22}\right))$$

## Values

 $$-1$$ $$1$$ $$3$$ $$5$$ $$7$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$19$$ $$21$$ $$1$$ $$1$$ $$e\left(\frac{8}{11}\right)$$ $$e\left(\frac{6}{11}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{9}{22}\right)$$ $$e\left(\frac{3}{22}\right)$$ $$e\left(\frac{3}{11}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{1}{11}\right)$$
 value at e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 184 }(51,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{184}(51,\cdot)) = \sum_{r\in \Z/184\Z} \chi_{184}(51,r) e\left(\frac{r}{92}\right) = 0.0$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 184 }(51,·),\chi_{ 184 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{184}(51,\cdot),\chi_{184}(1,\cdot)) = \sum_{r\in \Z/184\Z} \chi_{184}(51,r) \chi_{184}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 184 }(51,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{184}(51,·)) = \sum_{r \in \Z/184\Z} \chi_{184}(51,r) e\left(\frac{1 r + 2 r^{-1}}{184}\right) = 20.069545117+5.8929501476i$$