Properties

Label 569.17
Modulus $569$
Conductor $569$
Order $284$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(569, base_ring=CyclotomicField(284))
 
M = H._module
 
chi = DirichletCharacter(H, M([15]))
 
pari: [g,chi] = znchar(Mod(17,569))
 

Basic properties

Modulus: \(569\)
Conductor: \(569\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(284\)
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 569.g

\(\chi_{569}(2,\cdot)\) \(\chi_{569}(7,\cdot)\) \(\chi_{569}(8,\cdot)\) \(\chi_{569}(9,\cdot)\) \(\chi_{569}(10,\cdot)\) \(\chi_{569}(13,\cdot)\) \(\chi_{569}(17,\cdot)\) \(\chi_{569}(28,\cdot)\) \(\chi_{569}(32,\cdot)\) \(\chi_{569}(35,\cdot)\) \(\chi_{569}(36,\cdot)\) \(\chi_{569}(40,\cdot)\) \(\chi_{569}(45,\cdot)\) \(\chi_{569}(50,\cdot)\) \(\chi_{569}(52,\cdot)\) \(\chi_{569}(57,\cdot)\) \(\chi_{569}(61,\cdot)\) \(\chi_{569}(65,\cdot)\) \(\chi_{569}(66,\cdot)\) \(\chi_{569}(67,\cdot)\) \(\chi_{569}(68,\cdot)\) \(\chi_{569}(71,\cdot)\) \(\chi_{569}(79,\cdot)\) \(\chi_{569}(82,\cdot)\) \(\chi_{569}(85,\cdot)\) \(\chi_{569}(98,\cdot)\) \(\chi_{569}(112,\cdot)\) \(\chi_{569}(121,\cdot)\) \(\chi_{569}(126,\cdot)\) \(\chi_{569}(127,\cdot)\) ...

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: $\Q(\zeta_{284})$
Fixed field: Number field defined by a degree 284 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{15}{284}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 569 }(17, a) \) \(1\)\(1\)\(e\left(\frac{133}{142}\right)\)\(e\left(\frac{15}{284}\right)\)\(e\left(\frac{62}{71}\right)\)\(e\left(\frac{45}{71}\right)\)\(e\left(\frac{281}{284}\right)\)\(e\left(\frac{103}{142}\right)\)\(e\left(\frac{115}{142}\right)\)\(e\left(\frac{15}{142}\right)\)\(e\left(\frac{81}{142}\right)\)\(e\left(\frac{49}{284}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 569 }(17,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 569 }(17,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 569 }(17,·),\chi_{ 569 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 569 }(17,·)) \;\) at \(\; a,b = \) e.g. 1,2