Properties

Label 847.439
Modulus $847$
Conductor $847$
Order $66$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(847, base_ring=CyclotomicField(66))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([55,51]))
 
pari: [g,chi] = znchar(Mod(439,847))
 

Basic properties

Modulus: \(847\)
Conductor: \(847\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
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.x

\(\chi_{847}(10,\cdot)\) \(\chi_{847}(54,\cdot)\) \(\chi_{847}(87,\cdot)\) \(\chi_{847}(131,\cdot)\) \(\chi_{847}(164,\cdot)\) \(\chi_{847}(208,\cdot)\) \(\chi_{847}(285,\cdot)\) \(\chi_{847}(318,\cdot)\) \(\chi_{847}(395,\cdot)\) \(\chi_{847}(439,\cdot)\) \(\chi_{847}(472,\cdot)\) \(\chi_{847}(516,\cdot)\) \(\chi_{847}(549,\cdot)\) \(\chi_{847}(593,\cdot)\) \(\chi_{847}(626,\cdot)\) \(\chi_{847}(670,\cdot)\) \(\chi_{847}(703,\cdot)\) \(\chi_{847}(747,\cdot)\) \(\chi_{847}(780,\cdot)\) \(\chi_{847}(824,\cdot)\)

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: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 66 polynomial

Values on generators

\((122,365)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{17}{22}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(12\)\(13\)
\(1\)\(1\)\(e\left(\frac{29}{66}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{29}{33}\right)\)\(e\left(\frac{23}{66}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{26}{33}\right)\)\(e\left(\frac{47}{66}\right)\)\(e\left(\frac{6}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 847 }(439,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 847 }(439,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 847 }(439,·),\chi_{ 847 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 847 }(439,·)) \;\) at \(\; a,b = \) e.g. 1,2