# Properties

 Label 192.41 Modulus $192$ Conductor $96$ Order $8$ Real no Primitive no Minimal no Parity odd

# Related objects

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

sage: H = DirichletGroup(192)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,7,4]))

pari: [g,chi] = znchar(Mod(41,192))

## Basic properties

 Modulus: $$192$$ Conductor: $$96$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$8$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{96}(77,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: no Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 192.p

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

$$(127,133,65)$$ → $$(1,e\left(\frac{7}{8}\right),-1)$$

## Values

 $$-1$$ $$1$$ $$5$$ $$7$$ $$11$$ $$13$$ $$17$$ $$19$$ $$23$$ $$25$$ $$29$$ $$31$$ $$-1$$ $$1$$ $$e\left(\frac{3}{8}\right)$$ $$-i$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{1}{8}\right)$$ $$1$$ $$e\left(\frac{1}{8}\right)$$ $$-i$$ $$-i$$ $$e\left(\frac{1}{8}\right)$$ $$1$$
 value at e.g. 2

## Related number fields

 Field of values: $$\Q(\zeta_{8})$$ Fixed field: 8.0.173946175488.1

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 192 }(41,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{192}(41,\cdot)) = \sum_{r\in \Z/192\Z} \chi_{192}(41,r) e\left(\frac{r}{96}\right) = 3.8229739416+19.2193878738i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 192 }(41,·),\chi_{ 192 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{192}(41,\cdot),\chi_{192}(1,\cdot)) = \sum_{r\in \Z/192\Z} \chi_{192}(41,r) \chi_{192}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 192 }(41,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{192}(41,·)) = \sum_{r \in \Z/192\Z} \chi_{192}(41,r) e\left(\frac{1 r + 2 r^{-1}}{192}\right) = 0.0$$