# Properties

 Modulus 91 Conductor 91 Order 12 Real no Primitive yes Minimal yes Parity odd Orbit label 91.bd

# Related objects

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

sage: H = DirichletGroup(91)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([8,1]))

pari: [g,chi] = znchar(Mod(67,91))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Modulus = 91 Conductor = 91 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 12 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 = 91.bd Orbit index = 30

## 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

$$(66,15)$$ → $$(e\left(\frac{2}{3}\right),e\left(\frac{1}{12}\right))$$

## Values

 -1 1 2 3 4 5 6 8 9 10 11 12 $$-1$$ $$1$$ $$e\left(\frac{5}{12}\right)$$ $$1$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{12}\right)$$ $$e\left(\frac{5}{12}\right)$$ $$i$$ $$1$$ $$-1$$ $$i$$ $$e\left(\frac{5}{6}\right)$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{12})$$

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 91 }(67,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{91}(67,\cdot)) = \sum_{r\in \Z/91\Z} \chi_{91}(67,r) e\left(\frac{2r}{91}\right) = 9.5014959181+-0.8494558959i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 91 }(67,·),\chi_{ 91 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{91}(67,\cdot),\chi_{91}(1,\cdot)) = \sum_{r\in \Z/91\Z} \chi_{91}(67,r) \chi_{91}(1,1-r) = 1$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 91 }(67,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{91}(67,·)) = \sum_{r \in \Z/91\Z} \chi_{91}(67,r) e\left(\frac{1 r + 2 r^{-1}}{91}\right) = -8.0920782671+2.1682658368i$$