Properties

Label 571.29
Modulus $571$
Conductor $571$
Order $57$
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(114))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([40]))
 
pari: [g,chi] = znchar(Mod(29,571))
 

Basic properties

Modulus: \(571\)
Conductor: \(571\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(57\)
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.k

\(\chi_{571}(4,\cdot)\) \(\chi_{571}(16,\cdot)\) \(\chi_{571}(29,\cdot)\) \(\chi_{571}(82,\cdot)\) \(\chi_{571}(124,\cdot)\) \(\chi_{571}(143,\cdot)\) \(\chi_{571}(150,\cdot)\) \(\chi_{571}(220,\cdot)\) \(\chi_{571}(231,\cdot)\) \(\chi_{571}(236,\cdot)\) \(\chi_{571}(256,\cdot)\) \(\chi_{571}(258,\cdot)\) \(\chi_{571}(270,\cdot)\) \(\chi_{571}(285,\cdot)\) \(\chi_{571}(309,\cdot)\) \(\chi_{571}(328,\cdot)\) \(\chi_{571}(339,\cdot)\) \(\chi_{571}(362,\cdot)\) \(\chi_{571}(373,\cdot)\) \(\chi_{571}(376,\cdot)\) \(\chi_{571}(383,\cdot)\) \(\chi_{571}(396,\cdot)\) \(\chi_{571}(418,\cdot)\) \(\chi_{571}(436,\cdot)\) \(\chi_{571}(442,\cdot)\) \(\chi_{571}(443,\cdot)\) \(\chi_{571}(453,\cdot)\) \(\chi_{571}(464,\cdot)\) \(\chi_{571}(486,\cdot)\) \(\chi_{571}(496,\cdot)\) ...

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: $\Q(\zeta_{57})$
Fixed field: Number field defined by a degree 57 polynomial

Values on generators

\(3\) → \(e\left(\frac{20}{57}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{13}{57}\right)\)\(e\left(\frac{20}{57}\right)\)\(e\left(\frac{26}{57}\right)\)\(e\left(\frac{4}{57}\right)\)\(e\left(\frac{11}{19}\right)\)\(e\left(\frac{1}{19}\right)\)\(e\left(\frac{13}{19}\right)\)\(e\left(\frac{40}{57}\right)\)\(e\left(\frac{17}{57}\right)\)\(e\left(\frac{2}{57}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 571 }(29,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 571 }(29,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 571 }(29,·),\chi_{ 571 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 571 }(29,·)) \;\) at \(\; a,b = \) e.g. 1,2