# Properties

 Label 128.113 Modulus $128$ Conductor $32$ Order $8$ Real no Primitive no Minimal no Parity even

# Learn more about

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

sage: H = DirichletGroup(128)

sage: M = H._module

sage: chi = DirichletCharacter(H, M([0,1]))

pari: [g,chi] = znchar(Mod(113,128))

## Basic properties

 Modulus: $$128$$ Conductor: $$32$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$8$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{32}(5,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: no Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 128.g

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

$$(127,5)$$ → $$(1,e\left(\frac{1}{8}\right))$$

## Values

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

## Related number fields

 Field of values: $$\Q(\zeta_{8})$$ Fixed field: $$\Q(\zeta_{32})^+$$

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 128 }(113,·) )\;$$ at $$\;a =$$ e.g. 2
$$\displaystyle \tau_{2}(\chi_{128}(113,\cdot)) = \sum_{r\in \Z/128\Z} \chi_{128}(113,r) e\left(\frac{r}{64}\right) = 0.0$$

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 128 }(113,·),\chi_{ 128 }(n,·)) \;$$ for $$\; n =$$ e.g. 1
$$\displaystyle J(\chi_{128}(113,\cdot),\chi_{128}(1,\cdot)) = \sum_{r\in \Z/128\Z} \chi_{128}(113,r) \chi_{128}(1,1-r) = 0$$

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 128 }(113,·)) \;$$ at $$\; a,b =$$ e.g. 1,2
$$\displaystyle K(1,2,\chi_{128}(113,·)) = \sum_{r \in \Z/128\Z} \chi_{128}(113,r) e\left(\frac{1 r + 2 r^{-1}}{128}\right) = 0.0$$