Properties

Label 569.20
Modulus $569$
Conductor $569$
Order $142$
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([45]))
 
pari: [g,chi] = znchar(Mod(20,569))
 

Basic properties

Modulus: \(569\)
Conductor: \(569\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(142\)
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.f

\(\chi_{569}(4,\cdot)\) \(\chi_{569}(14,\cdot)\) \(\chi_{569}(20,\cdot)\) \(\chi_{569}(26,\cdot)\) \(\chi_{569}(34,\cdot)\) \(\chi_{569}(41,\cdot)\) \(\chi_{569}(43,\cdot)\) \(\chi_{569}(49,\cdot)\) \(\chi_{569}(64,\cdot)\) \(\chi_{569}(70,\cdot)\) \(\chi_{569}(72,\cdot)\) \(\chi_{569}(81,\cdot)\) \(\chi_{569}(87,\cdot)\) \(\chi_{569}(91,\cdot)\) \(\chi_{569}(93,\cdot)\) \(\chi_{569}(100,\cdot)\) \(\chi_{569}(119,\cdot)\) \(\chi_{569}(122,\cdot)\) \(\chi_{569}(130,\cdot)\) \(\chi_{569}(132,\cdot)\) \(\chi_{569}(158,\cdot)\) \(\chi_{569}(169,\cdot)\) \(\chi_{569}(170,\cdot)\) \(\chi_{569}(197,\cdot)\) \(\chi_{569}(205,\cdot)\) \(\chi_{569}(215,\cdot)\) \(\chi_{569}(221,\cdot)\) \(\chi_{569}(224,\cdot)\) \(\chi_{569}(242,\cdot)\) \(\chi_{569}(245,\cdot)\) ...

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 142 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{45}{142}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 569 }(20, a) \) \(1\)\(1\)\(e\left(\frac{44}{71}\right)\)\(e\left(\frac{45}{142}\right)\)\(e\left(\frac{17}{71}\right)\)\(e\left(\frac{57}{71}\right)\)\(e\left(\frac{133}{142}\right)\)\(e\left(\frac{25}{71}\right)\)\(e\left(\frac{61}{71}\right)\)\(e\left(\frac{45}{71}\right)\)\(e\left(\frac{30}{71}\right)\)\(e\left(\frac{5}{142}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 569 }(20,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 569 }(20,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 569 }(20,·),\chi_{ 569 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 569 }(20,·)) \;\) at \(\; a,b = \) e.g. 1,2