Properties

Label 571.86
Modulus $571$
Conductor $571$
Order $190$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(571, base_ring=CyclotomicField(190))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([143]))
 
pari: [g,chi] = znchar(Mod(86,571))
 

Basic properties

Modulus: \(571\)
Conductor: \(571\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(190\)
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 571.n

\(\chi_{571}(7,\cdot)\) \(\chi_{571}(15,\cdot)\) \(\chi_{571}(22,\cdot)\) \(\chi_{571}(26,\cdot)\) \(\chi_{571}(27,\cdot)\) \(\chi_{571}(42,\cdot)\) \(\chi_{571}(48,\cdot)\) \(\chi_{571}(50,\cdot)\) \(\chi_{571}(68,\cdot)\) \(\chi_{571}(86,\cdot)\) \(\chi_{571}(87,\cdot)\) \(\chi_{571}(95,\cdot)\) \(\chi_{571}(111,\cdot)\) \(\chi_{571}(122,\cdot)\) \(\chi_{571}(132,\cdot)\) \(\chi_{571}(140,\cdot)\) \(\chi_{571}(156,\cdot)\) \(\chi_{571}(160,\cdot)\) \(\chi_{571}(161,\cdot)\) \(\chi_{571}(162,\cdot)\) \(\chi_{571}(166,\cdot)\) \(\chi_{571}(171,\cdot)\) \(\chi_{571}(187,\cdot)\) \(\chi_{571}(194,\cdot)\) \(\chi_{571}(217,\cdot)\) \(\chi_{571}(235,\cdot)\) \(\chi_{571}(241,\cdot)\) \(\chi_{571}(252,\cdot)\) \(\chi_{571}(254,\cdot)\) \(\chi_{571}(266,\cdot)\) ...

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: $\Q(\zeta_{95})$
Fixed field: Number field defined by a degree 190 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{143}{190}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(-1\)\(1\)\(e\left(\frac{7}{38}\right)\)\(e\left(\frac{143}{190}\right)\)\(e\left(\frac{7}{19}\right)\)\(e\left(\frac{58}{95}\right)\)\(e\left(\frac{89}{95}\right)\)\(e\left(\frac{163}{190}\right)\)\(e\left(\frac{21}{38}\right)\)\(e\left(\frac{48}{95}\right)\)\(e\left(\frac{151}{190}\right)\)\(e\left(\frac{67}{95}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 571 }(86,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 571 }(86,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 571 }(86,·),\chi_{ 571 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 571 }(86,·)) \;\) at \(\; a,b = \) e.g. 1,2