Properties

Label 847.114
Modulus $847$
Conductor $847$
Order $165$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(847, base_ring=CyclotomicField(330))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([110,186]))
 
pari: [g,chi] = znchar(Mod(114,847))
 

Basic properties

Modulus: \(847\)
Conductor: \(847\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(165\)
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 847.bc

\(\chi_{847}(4,\cdot)\) \(\chi_{847}(16,\cdot)\) \(\chi_{847}(25,\cdot)\) \(\chi_{847}(37,\cdot)\) \(\chi_{847}(53,\cdot)\) \(\chi_{847}(58,\cdot)\) \(\chi_{847}(60,\cdot)\) \(\chi_{847}(86,\cdot)\) \(\chi_{847}(93,\cdot)\) \(\chi_{847}(102,\cdot)\) \(\chi_{847}(114,\cdot)\) \(\chi_{847}(135,\cdot)\) \(\chi_{847}(137,\cdot)\) \(\chi_{847}(158,\cdot)\) \(\chi_{847}(163,\cdot)\) \(\chi_{847}(170,\cdot)\) \(\chi_{847}(179,\cdot)\) \(\chi_{847}(191,\cdot)\) \(\chi_{847}(207,\cdot)\) \(\chi_{847}(212,\cdot)\) \(\chi_{847}(214,\cdot)\) \(\chi_{847}(235,\cdot)\) \(\chi_{847}(240,\cdot)\) \(\chi_{847}(247,\cdot)\) \(\chi_{847}(256,\cdot)\) \(\chi_{847}(268,\cdot)\) \(\chi_{847}(284,\cdot)\) \(\chi_{847}(289,\cdot)\) \(\chi_{847}(291,\cdot)\) \(\chi_{847}(312,\cdot)\) ...

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: $\Q(\zeta_{165})$
Fixed field: Number field defined by a degree 165 polynomial (not computed)

Values on generators

\((122,365)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{31}{55}\right))\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(12\)\(13\)
\( \chi_{ 847 }(114, a) \) \(1\)\(1\)\(e\left(\frac{38}{165}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{76}{165}\right)\)\(e\left(\frac{62}{165}\right)\)\(e\left(\frac{9}{55}\right)\)\(e\left(\frac{38}{55}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{20}{33}\right)\)\(e\left(\frac{13}{33}\right)\)\(e\left(\frac{51}{55}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 847 }(114,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 847 }(114,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 847 }(114,·),\chi_{ 847 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 847 }(114,·)) \;\) at \(\; a,b = \) e.g. 1,2