# Properties

 Label 668.55 Modulus $668$ Conductor $668$ Order $166$ Real no Primitive yes Minimal yes Parity even

# Related objects

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

sage: H = DirichletGroup(668)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([83,29]))

pari: [g,chi] = znchar(Mod(55,668))

## Basic properties

 Modulus: $$668$$ Conductor: $$668$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$166$$ 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 668.h

sage: chi.galois_orbit()

pari: order = charorder(g,chi)

pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]

## Values on generators

$$(335,5)$$ → $$(-1,e\left(\frac{29}{166}\right))$$

## Values

 $$-1$$ $$1$$ $$3$$ $$5$$ $$7$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$19$$ $$21$$ $$1$$ $$1$$ $$e\left(\frac{153}{166}\right)$$ $$e\left(\frac{29}{166}\right)$$ $$e\left(\frac{19}{166}\right)$$ $$e\left(\frac{70}{83}\right)$$ $$e\left(\frac{65}{166}\right)$$ $$e\left(\frac{165}{166}\right)$$ $$e\left(\frac{8}{83}\right)$$ $$e\left(\frac{43}{166}\right)$$ $$e\left(\frac{105}{166}\right)$$ $$e\left(\frac{3}{83}\right)$$
 value at e.g. 2

## Related number fields

 Field of values: $\Q(\zeta_{83})$ Fixed field: Number field defined by a degree 166 polynomial

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 668 }(55,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{668}(55,\cdot)) = \sum_{r\in \Z/668\Z} \chi_{668}(55,r) e\left(\frac{r}{334}\right) = -0.0$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 668 }(55,·),\chi_{ 668 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{668}(55,\cdot),\chi_{668}(1,\cdot)) = \sum_{r\in \Z/668\Z} \chi_{668}(55,r) \chi_{668}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 668 }(55,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{668}(55,·)) = \sum_{r \in \Z/668\Z} \chi_{668}(55,r) e\left(\frac{1 r + 2 r^{-1}}{668}\right) = 6.8149576641+-0.2580729053i$$