Properties

Label 847.24
Modulus $847$
Conductor $847$
Order $330$
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(330))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([55,273]))
 
pari: [g,chi] = znchar(Mod(24,847))
 

Basic properties

Modulus: \(847\)
Conductor: \(847\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(330\)
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.be

\(\chi_{847}(17,\cdot)\) \(\chi_{847}(19,\cdot)\) \(\chi_{847}(24,\cdot)\) \(\chi_{847}(52,\cdot)\) \(\chi_{847}(61,\cdot)\) \(\chi_{847}(68,\cdot)\) \(\chi_{847}(73,\cdot)\) \(\chi_{847}(96,\cdot)\) \(\chi_{847}(101,\cdot)\) \(\chi_{847}(117,\cdot)\) \(\chi_{847}(129,\cdot)\) \(\chi_{847}(138,\cdot)\) \(\chi_{847}(145,\cdot)\) \(\chi_{847}(150,\cdot)\) \(\chi_{847}(171,\cdot)\) \(\chi_{847}(173,\cdot)\) \(\chi_{847}(178,\cdot)\) \(\chi_{847}(194,\cdot)\) \(\chi_{847}(206,\cdot)\) \(\chi_{847}(222,\cdot)\) \(\chi_{847}(227,\cdot)\) \(\chi_{847}(248,\cdot)\) \(\chi_{847}(250,\cdot)\) \(\chi_{847}(255,\cdot)\) \(\chi_{847}(271,\cdot)\) \(\chi_{847}(283,\cdot)\) \(\chi_{847}(292,\cdot)\) \(\chi_{847}(299,\cdot)\) \(\chi_{847}(304,\cdot)\) \(\chi_{847}(325,\cdot)\) ...

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 330 polynomial (not computed)

Values on generators

\((122,365)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{91}{110}\right))\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(12\)\(13\)
\(1\)\(1\)\(e\left(\frac{53}{330}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{53}{165}\right)\)\(e\left(\frac{17}{330}\right)\)\(e\left(\frac{7}{55}\right)\)\(e\left(\frac{53}{110}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{7}{33}\right)\)\(e\left(\frac{19}{66}\right)\)\(e\left(\frac{3}{55}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 847 }(24,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 847 }(24,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 847 }(24,·),\chi_{ 847 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 847 }(24,·)) \;\) at \(\; a,b = \) e.g. 1,2