Properties

Label 589.283
Modulus $589$
Conductor $589$
Order $45$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(589, base_ring=CyclotomicField(90))
 
M = H._module
 
chi = DirichletCharacter(H, M([50,54]))
 
pari: [g,chi] = znchar(Mod(283,589))
 

Basic properties

Modulus: \(589\)
Conductor: \(589\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(45\)
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 589.cb

\(\chi_{589}(4,\cdot)\) \(\chi_{589}(16,\cdot)\) \(\chi_{589}(35,\cdot)\) \(\chi_{589}(47,\cdot)\) \(\chi_{589}(66,\cdot)\) \(\chi_{589}(101,\cdot)\) \(\chi_{589}(157,\cdot)\) \(\chi_{589}(188,\cdot)\) \(\chi_{589}(194,\cdot)\) \(\chi_{589}(225,\cdot)\) \(\chi_{589}(233,\cdot)\) \(\chi_{589}(252,\cdot)\) \(\chi_{589}(256,\cdot)\) \(\chi_{589}(264,\cdot)\) \(\chi_{589}(283,\cdot)\) \(\chi_{589}(405,\cdot)\) \(\chi_{589}(442,\cdot)\) \(\chi_{589}(473,\cdot)\) \(\chi_{589}(481,\cdot)\) \(\chi_{589}(498,\cdot)\) \(\chi_{589}(500,\cdot)\) \(\chi_{589}(529,\cdot)\) \(\chi_{589}(560,\cdot)\) \(\chi_{589}(574,\cdot)\)

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: $\Q(\zeta_{45})$
Fixed field: Number field defined by a degree 45 polynomial

Values on generators

\((249,96)\) → \((e\left(\frac{5}{9}\right),e\left(\frac{3}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 589 }(283, a) \) \(1\)\(1\)\(e\left(\frac{43}{45}\right)\)\(e\left(\frac{37}{45}\right)\)\(e\left(\frac{41}{45}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{29}{45}\right)\)\(e\left(\frac{38}{45}\right)\)\(e\left(\frac{7}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 589 }(283,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 589 }(283,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 589 }(283,·),\chi_{ 589 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 589 }(283,·)) \;\) at \(\; a,b = \) e.g. 1,2