# Properties

 Label 283.3 Modulus $283$ Conductor $283$ Order $282$ 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(283)

sage: M = H._module

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

pari: [g,chi] = znchar(Mod(3,283))

## Basic properties

 Modulus: $$283$$ Conductor: $$283$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$282$$ 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 283.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

$$3$$ → $$e\left(\frac{1}{282}\right)$$

## Values

 $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$7$$ $$8$$ $$9$$ $$10$$ $$11$$ $$-1$$ $$1$$ $$e\left(\frac{53}{94}\right)$$ $$e\left(\frac{1}{282}\right)$$ $$e\left(\frac{6}{47}\right)$$ $$e\left(\frac{233}{282}\right)$$ $$e\left(\frac{80}{141}\right)$$ $$e\left(\frac{139}{141}\right)$$ $$e\left(\frac{65}{94}\right)$$ $$e\left(\frac{1}{141}\right)$$ $$e\left(\frac{55}{141}\right)$$ $$e\left(\frac{76}{141}\right)$$
 value at e.g. 2

## Related number fields

 Field of values: $\Q(\zeta_{141})$ Fixed field: Number field defined by a degree 282 polynomial

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 283 }(3,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{283}(3,\cdot)) = \sum_{r\in \Z/283\Z} \chi_{283}(3,r) e\left(\frac{2r}{283}\right) = -13.778406818+9.651709981i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 283 }(3,·),\chi_{ 283 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{283}(3,\cdot),\chi_{283}(1,\cdot)) = \sum_{r\in \Z/283\Z} \chi_{283}(3,r) \chi_{283}(1,1-r) = -1$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 283 }(3,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{283}(3,·)) = \sum_{r \in \Z/283\Z} \chi_{283}(3,r) e\left(\frac{1 r + 2 r^{-1}}{283}\right) = 10.350948622+2.1039229262i$$