# Properties

 Label 768.41 Modulus $768$ Conductor $384$ Order $32$ 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(768, base_ring=CyclotomicField(32))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,31,16]))

pari: [g,chi] = znchar(Mod(41,768))

## Basic properties

 Modulus: $$768$$ Conductor: $$384$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$32$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{384}(77,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: no Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 768.x

sage: chi.galois_orbit()

pari: order = charorder(g,chi)

pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]

## Related number fields

 Field of values: $$\Q(\zeta_{32})$$ Fixed field: 32.0.135104323545903136978453058557785670637514001130337144105502507008.1

## Values on generators

$$(511,517,257)$$ → $$(1,e\left(\frac{31}{32}\right),-1)$$

## Values

 $$-1$$ $$1$$ $$5$$ $$7$$ $$11$$ $$13$$ $$17$$ $$19$$ $$23$$ $$25$$ $$29$$ $$31$$ $$-1$$ $$1$$ $$e\left(\frac{15}{32}\right)$$ $$e\left(\frac{11}{16}\right)$$ $$e\left(\frac{27}{32}\right)$$ $$e\left(\frac{17}{32}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{9}{32}\right)$$ $$e\left(\frac{1}{16}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{21}{32}\right)$$ $$-i$$
 value at e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 768 }(41,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{768}(41,\cdot)) = \sum_{r\in \Z/768\Z} \chi_{768}(41,r) e\left(\frac{r}{384}\right) = -38.7676434394+-5.7506366741i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 768 }(41,·),\chi_{ 768 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{768}(41,\cdot),\chi_{768}(1,\cdot)) = \sum_{r\in \Z/768\Z} \chi_{768}(41,r) \chi_{768}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 768 }(41,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{768}(41,·)) = \sum_{r \in \Z/768\Z} \chi_{768}(41,r) e\left(\frac{1 r + 2 r^{-1}}{768}\right) = 0.0$$