Properties

Label 569.136
Modulus $569$
Conductor $569$
Order $71$
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(142))
 
M = H._module
 
chi = DirichletCharacter(H, M([106]))
 
pari: [g,chi] = znchar(Mod(136,569))
 

Basic properties

Modulus: \(569\)
Conductor: \(569\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(71\)
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.e

\(\chi_{569}(5,\cdot)\) \(\chi_{569}(16,\cdot)\) \(\chi_{569}(18,\cdot)\) \(\chi_{569}(25,\cdot)\) \(\chi_{569}(33,\cdot)\) \(\chi_{569}(56,\cdot)\) \(\chi_{569}(63,\cdot)\) \(\chi_{569}(69,\cdot)\) \(\chi_{569}(80,\cdot)\) \(\chi_{569}(90,\cdot)\) \(\chi_{569}(101,\cdot)\) \(\chi_{569}(104,\cdot)\) \(\chi_{569}(107,\cdot)\) \(\chi_{569}(111,\cdot)\) \(\chi_{569}(113,\cdot)\) \(\chi_{569}(114,\cdot)\) \(\chi_{569}(117,\cdot)\) \(\chi_{569}(125,\cdot)\) \(\chi_{569}(134,\cdot)\) \(\chi_{569}(136,\cdot)\) \(\chi_{569}(141,\cdot)\) \(\chi_{569}(142,\cdot)\) \(\chi_{569}(153,\cdot)\) \(\chi_{569}(164,\cdot)\) \(\chi_{569}(165,\cdot)\) \(\chi_{569}(172,\cdot)\) \(\chi_{569}(196,\cdot)\) \(\chi_{569}(209,\cdot)\) \(\chi_{569}(219,\cdot)\) \(\chi_{569}(249,\cdot)\) ...

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: $\Q(\zeta_{71})$
Fixed field: Number field defined by a degree 71 polynomial

Values on generators

\(3\) → \(e\left(\frac{53}{71}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 569 }(136, a) \) \(1\)\(1\)\(e\left(\frac{50}{71}\right)\)\(e\left(\frac{53}{71}\right)\)\(e\left(\frac{29}{71}\right)\)\(e\left(\frac{68}{71}\right)\)\(e\left(\frac{32}{71}\right)\)\(e\left(\frac{51}{71}\right)\)\(e\left(\frac{8}{71}\right)\)\(e\left(\frac{35}{71}\right)\)\(e\left(\frac{47}{71}\right)\)\(e\left(\frac{69}{71}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 569 }(136,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 569 }(136,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 569 }(136,·),\chi_{ 569 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 569 }(136,·)) \;\) at \(\; a,b = \) e.g. 1,2