Properties

Label 571.227
Modulus $571$
Conductor $571$
Order $285$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(571, base_ring=CyclotomicField(570))
 
M = H._module
 
chi = DirichletCharacter(H, M([94]))
 
pari: [g,chi] = znchar(Mod(227,571))
 

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()
 
order = charorder(g,chi)
 
[ 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

\(3\) → \(e\left(\frac{47}{285}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 571 }(227, a) \) \(1\)\(1\)\(e\left(\frac{50}{57}\right)\)\(e\left(\frac{47}{285}\right)\)\(e\left(\frac{43}{57}\right)\)\(e\left(\frac{169}{285}\right)\)\(e\left(\frac{4}{95}\right)\)\(e\left(\frac{9}{95}\right)\)\(e\left(\frac{12}{19}\right)\)\(e\left(\frac{94}{285}\right)\)\(e\left(\frac{134}{285}\right)\)\(e\left(\frac{56}{285}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 571 }(227,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 571 }(227,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 571 }(227,·),\chi_{ 571 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 571 }(227,·)) \;\) at \(\; a,b = \) e.g. 1,2