# Properties

 Modulus 203 Conductor 203 Order 84 Real no Primitive yes Minimal yes Parity even Orbit label 203.x

# Related objects

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

sage: H = DirichletGroup(203)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([70,69]))

pari: [g,chi] = znchar(Mod(68,203))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Modulus = 203 Conductor = 203 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 84 Real = no sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization] Primitive = yes Minimal = yes sage: chi.is_odd()  pari: zncharisodd(g,chi) Parity = even Orbit label = 203.x Orbit index = 24

## Galois orbit

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

$$(59,176)$$ → $$(e\left(\frac{5}{6}\right),e\left(\frac{23}{28}\right))$$

## Values

 -1 1 2 3 4 5 6 8 9 10 11 12 $$1$$ $$1$$ $$e\left(\frac{41}{84}\right)$$ $$e\left(\frac{79}{84}\right)$$ $$e\left(\frac{41}{42}\right)$$ $$e\left(\frac{5}{21}\right)$$ $$e\left(\frac{3}{7}\right)$$ $$e\left(\frac{13}{28}\right)$$ $$e\left(\frac{37}{42}\right)$$ $$e\left(\frac{61}{84}\right)$$ $$e\left(\frac{73}{84}\right)$$ $$e\left(\frac{11}{12}\right)$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{84})$$

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 203 }(68,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{203}(68,\cdot)) = \sum_{r\in \Z/203\Z} \chi_{203}(68,r) e\left(\frac{2r}{203}\right) = -13.2745106355+-5.1756513974i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 203 }(68,·),\chi_{ 203 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{203}(68,\cdot),\chi_{203}(1,\cdot)) = \sum_{r\in \Z/203\Z} \chi_{203}(68,r) \chi_{203}(1,1-r) = 1$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 203 }(68,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{203}(68,·)) = \sum_{r \in \Z/203\Z} \chi_{203}(68,r) e\left(\frac{1 r + 2 r^{-1}}{203}\right) = 0.2270623836+6.0683699548i$$