Properties

Label 571.23
Modulus $571$
Conductor $571$
Order $95$
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(571, base_ring=CyclotomicField(190))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([82]))
 
pari: [g,chi] = znchar(Mod(23,571))
 

Basic properties

Modulus: \(571\)
Conductor: \(571\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(95\)
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 571.l

\(\chi_{571}(6,\cdot)\) \(\chi_{571}(20,\cdot)\) \(\chi_{571}(23,\cdot)\) \(\chi_{571}(36,\cdot)\) \(\chi_{571}(38,\cdot)\) \(\chi_{571}(49,\cdot)\) \(\chi_{571}(51,\cdot)\) \(\chi_{571}(56,\cdot)\) \(\chi_{571}(65,\cdot)\) \(\chi_{571}(105,\cdot)\) \(\chi_{571}(117,\cdot)\) \(\chi_{571}(120,\cdot)\) \(\chi_{571}(123,\cdot)\) \(\chi_{571}(125,\cdot)\) \(\chi_{571}(138,\cdot)\) \(\chi_{571}(142,\cdot)\) \(\chi_{571}(146,\cdot)\) \(\chi_{571}(148,\cdot)\) \(\chi_{571}(149,\cdot)\) \(\chi_{571}(154,\cdot)\) \(\chi_{571}(158,\cdot)\) \(\chi_{571}(163,\cdot)\) \(\chi_{571}(176,\cdot)\) \(\chi_{571}(179,\cdot)\) \(\chi_{571}(182,\cdot)\) \(\chi_{571}(186,\cdot)\) \(\chi_{571}(189,\cdot)\) \(\chi_{571}(201,\cdot)\) \(\chi_{571}(206,\cdot)\) \(\chi_{571}(208,\cdot)\) ...

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 95 polynomial

Values on generators

\(3\) → \(e\left(\frac{41}{95}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{4}{19}\right)\)\(e\left(\frac{41}{95}\right)\)\(e\left(\frac{8}{19}\right)\)\(e\left(\frac{12}{95}\right)\)\(e\left(\frac{61}{95}\right)\)\(e\left(\frac{66}{95}\right)\)\(e\left(\frac{12}{19}\right)\)\(e\left(\frac{82}{95}\right)\)\(e\left(\frac{32}{95}\right)\)\(e\left(\frac{63}{95}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 571 }(23,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 571 }(23,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 571 }(23,·),\chi_{ 571 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 571 }(23,·)) \;\) at \(\; a,b = \) e.g. 1,2