# Properties

 Modulus 94 Conductor 47 Order 46 Real no Primitive no Minimal yes Parity odd Orbit label 94.d

# Related objects

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

sage: H = DirichletGroup(94)

sage: M = H._module

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

pari: [g,chi] = znchar(Mod(5,94))

## Basic properties

 sage: chi.conductor()  pari: znconreyconductor(g,chi) Modulus = 94 Conductor = 47 sage: chi.multiplicative_order()  pari: charorder(g,chi) Order = 46 Real = no sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization] Primitive = no Minimal = yes sage: chi.is_odd()  pari: zncharisodd(g,chi) Parity = odd Orbit label = 94.d Orbit index = 4

## 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

$$5$$ → $$e\left(\frac{1}{46}\right)$$

## Values

 -1 1 3 5 7 9 11 13 15 17 19 21 $$-1$$ $$1$$ $$e\left(\frac{10}{23}\right)$$ $$e\left(\frac{1}{46}\right)$$ $$e\left(\frac{16}{23}\right)$$ $$e\left(\frac{20}{23}\right)$$ $$e\left(\frac{7}{46}\right)$$ $$e\left(\frac{11}{46}\right)$$ $$e\left(\frac{21}{46}\right)$$ $$e\left(\frac{8}{23}\right)$$ $$e\left(\frac{45}{46}\right)$$ $$e\left(\frac{3}{23}\right)$$
value at  e.g. 2

## Related number fields

 Field of values $$\Q(\zeta_{23})$$

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 94 }(5,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{94}(5,\cdot)) = \sum_{r\in \Z/94\Z} \chi_{94}(5,r) e\left(\frac{r}{47}\right) = -6.6250006718+1.7633394736i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 94 }(5,·),\chi_{ 94 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{94}(5,\cdot),\chi_{94}(1,\cdot)) = \sum_{r\in \Z/94\Z} \chi_{94}(5,r) \chi_{94}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 94 }(5,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{94}(5,·)) = \sum_{r \in \Z/94\Z} \chi_{94}(5,r) e\left(\frac{1 r + 2 r^{-1}}{94}\right) = -3.6932722156+1.3125892608i$$