Properties

 Label 693.100 Modulus $693$ Conductor $7$ Order $3$ Real no Primitive no Minimal yes Parity even

Related objects

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

H = DirichletGroup(693, base_ring=CyclotomicField(6))

M = H._module

chi = DirichletCharacter(H, M([0,2,0]))

pari: [g,chi] = znchar(Mod(100,693))

Basic properties

 Modulus: $$693$$ Conductor: $$7$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$3$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: no, induced from $$\chi_{7}(2,\cdot)$$ sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: even sage: chi.is_odd()  pari: zncharisodd(g,chi)

Galois orbit 693.i

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: $$\mathbb{Q}(\zeta_3)$$ Fixed field: $$\Q(\zeta_{7})^+$$

Values on generators

$$(155,199,442)$$ → $$(1,e\left(\frac{1}{3}\right),1)$$

First values

 $$a$$ $$-1$$ $$1$$ $$2$$ $$4$$ $$5$$ $$8$$ $$10$$ $$13$$ $$16$$ $$17$$ $$19$$ $$20$$ $$\chi_{ 693 }(100, a)$$ $$1$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$ $$e\left(\frac{1}{3}\right)$$ $$1$$ $$e\left(\frac{2}{3}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{2}{3}\right)$$ $$1$$
sage: chi.jacobi_sum(n)

$$\chi_{ 693 }(100,a) \;$$ at $$\;a =$$ e.g. 2

Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 693 }(100,·) )\;$$ at $$\;a =$$ e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 693 }(100,·),\chi_{ 693 }(n,·)) \;$$ for $$\; n =$$ e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 693 }(100,·)) \;$$ at $$\; a,b =$$ e.g. 1,2