Properties

Label 571.31
Modulus $571$
Conductor $571$
Order $19$
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(38))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([6]))
 
pari: [g,chi] = znchar(Mod(31,571))
 

Basic properties

Modulus: \(571\)
Conductor: \(571\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(19\)
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.h

\(\chi_{571}(31,\cdot)\) \(\chi_{571}(55,\cdot)\) \(\chi_{571}(59,\cdot)\) \(\chi_{571}(64,\cdot)\) \(\chi_{571}(94,\cdot)\) \(\chi_{571}(99,\cdot)\) \(\chi_{571}(116,\cdot)\) \(\chi_{571}(131,\cdot)\) \(\chi_{571}(170,\cdot)\) \(\chi_{571}(214,\cdot)\) \(\chi_{571}(271,\cdot)\) \(\chi_{571}(306,\cdot)\) \(\chi_{571}(323,\cdot)\) \(\chi_{571}(350,\cdot)\) \(\chi_{571}(353,\cdot)\) \(\chi_{571}(390,\cdot)\) \(\chi_{571}(407,\cdot)\) \(\chi_{571}(563,\cdot)\)

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: \(\Q(\zeta_{19})\)
Fixed field: 19.19.41634173570364661205169708858211372543325791407961.1

Values on generators

\(3\) → \(e\left(\frac{3}{19}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{1}{19}\right)\)\(e\left(\frac{3}{19}\right)\)\(e\left(\frac{2}{19}\right)\)\(e\left(\frac{12}{19}\right)\)\(e\left(\frac{4}{19}\right)\)\(e\left(\frac{9}{19}\right)\)\(e\left(\frac{3}{19}\right)\)\(e\left(\frac{6}{19}\right)\)\(e\left(\frac{13}{19}\right)\)\(e\left(\frac{6}{19}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 571 }(31,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 571 }(31,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 571 }(31,·),\chi_{ 571 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 571 }(31,·)) \;\) at \(\; a,b = \) e.g. 1,2