# Properties

 Modulus 92 Conductor 92 Order 22 Real no Primitive yes Minimal yes Parity odd Orbit label 92.g

# Related objects

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

sage: H = DirichletGroup(92)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([11,14]))

pari: [g,chi] = znchar(Mod(59,92))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Modulus = 92 Conductor = 92 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 22 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 = odd Orbit label = 92.g Orbit index = 7

## 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

$$(47,5)$$ → $$(-1,e\left(\frac{7}{11}\right))$$

## Values

 -1 1 3 5 7 9 11 13 15 17 19 21 $$-1$$ $$1$$ $$e\left(\frac{15}{22}\right)$$ $$e\left(\frac{7}{11}\right)$$ $$e\left(\frac{13}{22}\right)$$ $$e\left(\frac{4}{11}\right)$$ $$e\left(\frac{5}{22}\right)$$ $$e\left(\frac{10}{11}\right)$$ $$e\left(\frac{7}{22}\right)$$ $$e\left(\frac{5}{11}\right)$$ $$e\left(\frac{1}{22}\right)$$ $$e\left(\frac{3}{11}\right)$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{11})$$

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 92 }(59,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{92}(59,\cdot)) = \sum_{r\in \Z/92\Z} \chi_{92}(59,r) e\left(\frac{r}{46}\right) = 0.0$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 92 }(59,·),\chi_{ 92 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{92}(59,\cdot),\chi_{92}(1,\cdot)) = \sum_{r\in \Z/92\Z} \chi_{92}(59,r) \chi_{92}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 92 }(59,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{92}(59,·)) = \sum_{r \in \Z/92\Z} \chi_{92}(59,r) e\left(\frac{1 r + 2 r^{-1}}{92}\right) = -7.7861661446+6.7467513699i$$