# Properties

 Label 256.9 Modulus $256$ Conductor $128$ Order $32$ Real no Primitive no Minimal no Parity even

# Related objects

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

sage: H = DirichletGroup(256)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,3]))

pari: [g,chi] = znchar(Mod(9,256))

## Basic properties

 Modulus: $$256$$ Conductor: $$128$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$32$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{128}(125,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: no Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 256.k

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

$$(255,5)$$ → $$(1,e\left(\frac{3}{32}\right))$$

## Values

 $$-1$$ $$1$$ $$3$$ $$5$$ $$7$$ $$9$$ $$11$$ $$13$$ $$15$$ $$17$$ $$19$$ $$21$$ $$1$$ $$1$$ $$e\left(\frac{9}{32}\right)$$ $$e\left(\frac{3}{32}\right)$$ $$e\left(\frac{15}{16}\right)$$ $$e\left(\frac{9}{16}\right)$$ $$e\left(\frac{31}{32}\right)$$ $$e\left(\frac{13}{32}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$e\left(\frac{5}{32}\right)$$ $$e\left(\frac{7}{32}\right)$$
 value at e.g. 2

## Related number fields

 Field of values: $$\Q(\zeta_{32})$$ Fixed field: $$\Q(\zeta_{128})^+$$

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 256 }(9,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{256}(9,\cdot)) = \sum_{r\in \Z/256\Z} \chi_{256}(9,r) e\left(\frac{r}{128}\right) = 9.6744673887+20.4549426972i$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 256 }(9,·),\chi_{ 256 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{256}(9,\cdot),\chi_{256}(1,\cdot)) = \sum_{r\in \Z/256\Z} \chi_{256}(9,r) \chi_{256}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 256 }(9,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{256}(9,·)) = \sum_{r \in \Z/256\Z} \chi_{256}(9,r) e\left(\frac{1 r + 2 r^{-1}}{256}\right) = 0.0$$