Properties

Label 571.8
Modulus $571$
Conductor $571$
Order $38$
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(38))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([33]))
 
pari: [g,chi] = znchar(Mod(8,571))
 

Basic properties

Modulus: \(571\)
Conductor: \(571\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(38\)
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.j

\(\chi_{571}(8,\cdot)\) \(\chi_{571}(164,\cdot)\) \(\chi_{571}(181,\cdot)\) \(\chi_{571}(218,\cdot)\) \(\chi_{571}(221,\cdot)\) \(\chi_{571}(248,\cdot)\) \(\chi_{571}(265,\cdot)\) \(\chi_{571}(300,\cdot)\) \(\chi_{571}(357,\cdot)\) \(\chi_{571}(401,\cdot)\) \(\chi_{571}(440,\cdot)\) \(\chi_{571}(455,\cdot)\) \(\chi_{571}(472,\cdot)\) \(\chi_{571}(477,\cdot)\) \(\chi_{571}(507,\cdot)\) \(\chi_{571}(512,\cdot)\) \(\chi_{571}(516,\cdot)\) \(\chi_{571}(540,\cdot)\)

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: 38.0.989773917474620481291516877896927890204180311865757468744614075195570463990089402894625057329215364491.1

Values on generators

\(3\) → \(e\left(\frac{33}{38}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(-1\)\(1\)\(e\left(\frac{11}{38}\right)\)\(e\left(\frac{33}{38}\right)\)\(e\left(\frac{11}{19}\right)\)\(e\left(\frac{9}{19}\right)\)\(e\left(\frac{3}{19}\right)\)\(e\left(\frac{23}{38}\right)\)\(e\left(\frac{33}{38}\right)\)\(e\left(\frac{14}{19}\right)\)\(e\left(\frac{29}{38}\right)\)\(e\left(\frac{14}{19}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 571 }(8,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 571 }(8,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 571 }(8,·),\chi_{ 571 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 571 }(8,·)) \;\) at \(\; a,b = \) e.g. 1,2