Properties

Label 1033.129
Modulus $1033$
Conductor $1033$
Order $43$
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(86))
 
M = H._module
 
chi = DirichletCharacter(H, M([38]))
 
pari: [g,chi] = znchar(Mod(129,1033))
 

Basic properties

Modulus: \(1033\)
Conductor: \(1033\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(43\)
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.i

\(\chi_{1033}(64,\cdot)\) \(\chi_{1033}(113,\cdot)\) \(\chi_{1033}(115,\cdot)\) \(\chi_{1033}(129,\cdot)\) \(\chi_{1033}(263,\cdot)\) \(\chi_{1033}(288,\cdot)\) \(\chi_{1033}(296,\cdot)\) \(\chi_{1033}(299,\cdot)\) \(\chi_{1033}(300,\cdot)\) \(\chi_{1033}(304,\cdot)\) \(\chi_{1033}(317,\cdot)\) \(\chi_{1033}(335,\cdot)\) \(\chi_{1033}(336,\cdot)\) \(\chi_{1033}(350,\cdot)\) \(\chi_{1033}(373,\cdot)\) \(\chi_{1033}(392,\cdot)\) \(\chi_{1033}(411,\cdot)\) \(\chi_{1033}(419,\cdot)\) \(\chi_{1033}(479,\cdot)\) \(\chi_{1033}(521,\cdot)\) \(\chi_{1033}(542,\cdot)\) \(\chi_{1033}(563,\cdot)\) \(\chi_{1033}(599,\cdot)\) \(\chi_{1033}(606,\cdot)\) \(\chi_{1033}(661,\cdot)\) \(\chi_{1033}(667,\cdot)\) \(\chi_{1033}(699,\cdot)\) \(\chi_{1033}(707,\cdot)\) \(\chi_{1033}(731,\cdot)\) \(\chi_{1033}(780,\cdot)\) ...

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: $\Q(\zeta_{43})$
Fixed field: Number field defined by a degree 43 polynomial

Values on generators

\(5\) → \(e\left(\frac{19}{43}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1033 }(129, a) \) \(1\)\(1\)\(e\left(\frac{36}{43}\right)\)\(e\left(\frac{8}{43}\right)\)\(e\left(\frac{29}{43}\right)\)\(e\left(\frac{19}{43}\right)\)\(e\left(\frac{1}{43}\right)\)\(e\left(\frac{7}{43}\right)\)\(e\left(\frac{22}{43}\right)\)\(e\left(\frac{16}{43}\right)\)\(e\left(\frac{12}{43}\right)\)\(e\left(\frac{28}{43}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1033 }(129,a) \;\) at \(\;a = \) e.g. 2