# Properties

 Label 68.53 Modulus $68$ Conductor $17$ Order $8$ Real no Primitive no Minimal yes Parity even

# Related objects

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(68, base_ring=CyclotomicField(8))

M = H._module

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

pari: [g,chi] = znchar(Mod(53,68))

## Basic properties

 Modulus: $$68$$ Conductor: $$17$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$8$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{17}(2,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

## Galois orbit 68.h

sage: chi.galois_orbit()

order = charorder(g,chi)

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

## Related number fields

 Field of values: $$\Q(\zeta_{8})$$ Fixed field: $$\Q(\zeta_{17})^+$$

## Values on generators

$$(35,37)$$ → $$(1,e\left(\frac{7}{8}\right))$$

## Values

 $$a$$ $$-1$$ $$1$$ $$3$$ $$5$$ $$7$$ $$9$$ $$11$$ $$13$$ $$15$$ $$19$$ $$21$$ $$23$$ $$\chi_{ 68 }(53, a)$$ $$1$$ $$1$$ $$e\left(\frac{7}{8}\right)$$ $$e\left(\frac{3}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$ $$-i$$ $$e\left(\frac{1}{8}\right)$$ $$-1$$ $$i$$ $$i$$ $$-1$$ $$e\left(\frac{1}{8}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 68 }(53,a) \;$$ at $$\;a =$$ e.g. 2

## Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 68 }(53,·) )\;$$ at $$\;a =$$ e.g. 2

## Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 68 }(53,·),\chi_{ 68 }(n,·)) \;$$ for $$\; n =$$ e.g. 1

## Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 68 }(53,·)) \;$$ at $$\; a,b =$$ e.g. 1,2