# Properties

 Label 633.266 Modulus $633$ Conductor $633$ Order $10$ Real no Primitive yes Minimal yes Parity odd

# Related objects

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

sage: H = DirichletGroup(633)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([5,4]))

pari: [g,chi] = znchar(Mod(266,633))

## Basic properties

 Modulus: $$633$$ Conductor: $$633$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$10$$ 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 633.m

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

$$(212,424)$$ → $$(-1,e\left(\frac{2}{5}\right))$$

## Values

 $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$7$$ $$8$$ $$10$$ $$11$$ $$13$$ $$14$$ $$16$$ $$-1$$ $$1$$ $$e\left(\frac{9}{10}\right)$$ $$e\left(\frac{4}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$e\left(\frac{7}{10}\right)$$ $$e\left(\frac{1}{5}\right)$$ $$e\left(\frac{3}{10}\right)$$ $$e\left(\frac{3}{5}\right)$$ $$-1$$ $$e\left(\frac{3}{5}\right)$$
 value at e.g. 2

## Related number fields

 Field of values: $$\Q(\zeta_{5})$$ Fixed field: 10.0.954697787248807052883.2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 633 }(266,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{633}(266,\cdot)) = \sum_{r\in \Z/633\Z} \chi_{633}(266,r) e\left(\frac{2r}{633}\right) = -16.6082885625+18.8988028993i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 633 }(266,·),\chi_{ 633 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{633}(266,\cdot),\chi_{633}(1,\cdot)) = \sum_{r\in \Z/633\Z} \chi_{633}(266,r) \chi_{633}(1,1-r) = 1$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 633 }(266,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{633}(266,·)) = \sum_{r \in \Z/633\Z} \chi_{633}(266,r) e\left(\frac{1 r + 2 r^{-1}}{633}\right) = -0.0$$