Properties

Label 1033.18
Modulus $1033$
Conductor $1033$
Order $129$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1033, base_ring=CyclotomicField(258))
 
M = H._module
 
chi = DirichletCharacter(H, M([98]))
 
pari: [g,chi] = znchar(Mod(18,1033))
 

Basic properties

Modulus: \(1033\)
Conductor: \(1033\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(129\)
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 1033.k

\(\chi_{1033}(4,\cdot)\) \(\chi_{1033}(16,\cdot)\) \(\chi_{1033}(18,\cdot)\) \(\chi_{1033}(19,\cdot)\) \(\chi_{1033}(21,\cdot)\) \(\chi_{1033}(72,\cdot)\) \(\chi_{1033}(74,\cdot)\) \(\chi_{1033}(75,\cdot)\) \(\chi_{1033}(76,\cdot)\) \(\chi_{1033}(81,\cdot)\) \(\chi_{1033}(84,\cdot)\) \(\chi_{1033}(98,\cdot)\) \(\chi_{1033}(102,\cdot)\) \(\chi_{1033}(119,\cdot)\) \(\chi_{1033}(151,\cdot)\) \(\chi_{1033}(163,\cdot)\) \(\chi_{1033}(167,\cdot)\) \(\chi_{1033}(183,\cdot)\) \(\chi_{1033}(186,\cdot)\) \(\chi_{1033}(211,\cdot)\) \(\chi_{1033}(217,\cdot)\) \(\chi_{1033}(235,\cdot)\) \(\chi_{1033}(246,\cdot)\) \(\chi_{1033}(249,\cdot)\) \(\chi_{1033}(256,\cdot)\) \(\chi_{1033}(277,\cdot)\) \(\chi_{1033}(287,\cdot)\) \(\chi_{1033}(307,\cdot)\) \(\chi_{1033}(311,\cdot)\) \(\chi_{1033}(324,\cdot)\) ...

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: $\Q(\zeta_{129})$
Fixed field: Number field defined by a degree 129 polynomial (not computed)

Values on generators

\(5\) → \(e\left(\frac{49}{129}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1033 }(18, a) \) \(1\)\(1\)\(e\left(\frac{34}{129}\right)\)\(e\left(\frac{41}{129}\right)\)\(e\left(\frac{68}{129}\right)\)\(e\left(\frac{49}{129}\right)\)\(e\left(\frac{25}{43}\right)\)\(e\left(\frac{3}{43}\right)\)\(e\left(\frac{34}{43}\right)\)\(e\left(\frac{82}{129}\right)\)\(e\left(\frac{83}{129}\right)\)\(e\left(\frac{79}{129}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1033 }(18,a) \;\) at \(\;a = \) e.g. 2