Properties

Label 571.21
Modulus $571$
Conductor $571$
Order $285$
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(570))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([232]))
 
pari: [g,chi] = znchar(Mod(21,571))
 

Basic properties

Modulus: \(571\)
Conductor: \(571\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(285\)
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.o

\(\chi_{571}(5,\cdot)\) \(\chi_{571}(9,\cdot)\) \(\chi_{571}(11,\cdot)\) \(\chi_{571}(13,\cdot)\) \(\chi_{571}(14,\cdot)\) \(\chi_{571}(21,\cdot)\) \(\chi_{571}(24,\cdot)\) \(\chi_{571}(25,\cdot)\) \(\chi_{571}(30,\cdot)\) \(\chi_{571}(34,\cdot)\) \(\chi_{571}(37,\cdot)\) \(\chi_{571}(43,\cdot)\) \(\chi_{571}(44,\cdot)\) \(\chi_{571}(45,\cdot)\) \(\chi_{571}(52,\cdot)\) \(\chi_{571}(54,\cdot)\) \(\chi_{571}(57,\cdot)\) \(\chi_{571}(61,\cdot)\) \(\chi_{571}(66,\cdot)\) \(\chi_{571}(70,\cdot)\) \(\chi_{571}(78,\cdot)\) \(\chi_{571}(80,\cdot)\) \(\chi_{571}(81,\cdot)\) \(\chi_{571}(83,\cdot)\) \(\chi_{571}(84,\cdot)\) \(\chi_{571}(92,\cdot)\) \(\chi_{571}(96,\cdot)\) \(\chi_{571}(97,\cdot)\) \(\chi_{571}(100,\cdot)\) \(\chi_{571}(115,\cdot)\) ...

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: $\Q(\zeta_{285})$
Fixed field: Number field defined by a degree 285 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{116}{285}\right)\)

Values

\(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\(1\)\(1\)\(e\left(\frac{47}{57}\right)\)\(e\left(\frac{116}{285}\right)\)\(e\left(\frac{37}{57}\right)\)\(e\left(\frac{217}{285}\right)\)\(e\left(\frac{22}{95}\right)\)\(e\left(\frac{2}{95}\right)\)\(e\left(\frac{9}{19}\right)\)\(e\left(\frac{232}{285}\right)\)\(e\left(\frac{167}{285}\right)\)\(e\left(\frac{23}{285}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 571 }(21,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 571 }(21,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 571 }(21,·),\chi_{ 571 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 571 }(21,·)) \;\) at \(\; a,b = \) e.g. 1,2