Properties

 Label 287.167 Modulus $287$ Conductor $287$ Order $8$ Real no Primitive yes Minimal yes Parity even

Related objects

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

H = DirichletGroup(287, base_ring=CyclotomicField(8))

M = H._module

chi = DirichletCharacter(H, M([4,3]))

pari: [g,chi] = znchar(Mod(167,287))

Basic properties

 Modulus: $$287$$ Conductor: $$287$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$8$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: yes sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

Galois orbit 287.l

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: 8.8.467605011588281.1

Values on generators

$$(206,211)$$ → $$(-1,e\left(\frac{3}{8}\right))$$

First values

 $$a$$ $$-1$$ $$1$$ $$2$$ $$3$$ $$4$$ $$5$$ $$6$$ $$8$$ $$9$$ $$10$$ $$11$$ $$12$$ $$\chi_{ 287 }(167, a)$$ $$1$$ $$1$$ $$-i$$ $$e\left(\frac{1}{8}\right)$$ $$-1$$ $$-i$$ $$e\left(\frac{7}{8}\right)$$ $$i$$ $$i$$ $$-1$$ $$e\left(\frac{1}{8}\right)$$ $$e\left(\frac{5}{8}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 287 }(167,a) \;$$ at $$\;a =$$ e.g. 2

Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 287 }(167,·) )\;$$ at $$\;a =$$ e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 287 }(167,·),\chi_{ 287 }(n,·)) \;$$ for $$\; n =$$ e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 287 }(167,·)) \;$$ at $$\; a,b =$$ e.g. 1,2