# Properties

 Modulus 89 Conductor 89 Order 2 Real yes Primitive yes Minimal yes Parity even Orbit label 89.b

# Related objects

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

sage: H = DirichletGroup(89)

sage: M = H._module

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

pari: [g,chi] = znchar(Mod(88,89))

## Kronecker symbol representation

sage: kronecker_character(89)

pari: znchartokronecker(g,chi)

$$\displaystyle\left(\frac{89}{\bullet}\right)$$

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Modulus = 89 Conductor = 89 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 2 Real = yes 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 = 89.b Orbit index = 2

## 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

$$3$$ → $$-1$$

## Values

 -1 1 2 3 4 5 6 7 8 9 10 11 $$1$$ $$1$$ $$1$$ $$-1$$ $$1$$ $$1$$ $$-1$$ $$-1$$ $$1$$ $$1$$ $$1$$ $$1$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q$$

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 89 }(88,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{89}(88,\cdot)) = \sum_{r\in \Z/89\Z} \chi_{89}(88,r) e\left(\frac{2r}{89}\right) = 9.4339811321$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 89 }(88,·),\chi_{ 89 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{89}(88,\cdot),\chi_{89}(1,\cdot)) = \sum_{r\in \Z/89\Z} \chi_{89}(88,r) \chi_{89}(1,1-r) = -1$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 89 }(88,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{89}(88,·)) = \sum_{r \in \Z/89\Z} \chi_{89}(88,r) e\left(\frac{1 r + 2 r^{-1}}{89}\right) = -17.4634135413$$