Properties

 Label 456.59 Modulus $456$ Conductor $456$ Order $18$ Real no Primitive yes Minimal yes Parity odd

Related objects

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

sage: H = DirichletGroup(456, base_ring=CyclotomicField(18))

sage: M = H._module

sage: chi = DirichletCharacter(H, M([9,9,9,1]))

pari: [g,chi] = znchar(Mod(59,456))

Basic properties

 Modulus: $$456$$ Conductor: $$456$$ sage: chi.conductor()  pari: znconreyconductor(g,chi) Order: $$18$$ sage: chi.multiplicative_order()  pari: charorder(g,chi) Real: no Primitive: yes sage: chi.is_primitive()  pari: #znconreyconductor(g,chi)==1 Minimal: yes Parity: odd sage: chi.is_odd()  pari: zncharisodd(g,chi)

Galois orbit 456.bt

sage: chi.galois_orbit()

pari: order = charorder(g,chi)

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

Related number fields

 Field of values: $$\Q(\zeta_{9})$$ Fixed field: 18.0.14478127324240404768365927869710336.1

Values on generators

$$(343,229,305,97)$$ → $$(-1,-1,-1,e\left(\frac{1}{18}\right))$$

Values

 $$a$$ $$-1$$ $$1$$ $$5$$ $$7$$ $$11$$ $$13$$ $$17$$ $$23$$ $$25$$ $$29$$ $$31$$ $$35$$ $$\chi_{ 456 }(59, a)$$ $$-1$$ $$1$$ $$e\left(\frac{8}{9}\right)$$ $$e\left(\frac{5}{6}\right)$$ $$e\left(\frac{1}{6}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{1}{18}\right)$$ $$e\left(\frac{1}{9}\right)$$ $$e\left(\frac{7}{9}\right)$$ $$e\left(\frac{17}{18}\right)$$ $$e\left(\frac{1}{3}\right)$$ $$e\left(\frac{13}{18}\right)$$
sage: chi.jacobi_sum(n)

$$\chi_{ 456 }(59,a) \;$$ at $$\;a =$$ e.g. 2

Gauss sum

sage: chi.gauss_sum(a)

pari: znchargauss(g,chi,a)

$$\tau_{ a }( \chi_{ 456 }(59,·) )\;$$ at $$\;a =$$ e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)

$$J(\chi_{ 456 }(59,·),\chi_{ 456 }(n,·)) \;$$ for $$\; n =$$ e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)

$$K(a,b,\chi_{ 456 }(59,·)) \;$$ at $$\; a,b =$$ e.g. 1,2