Properties

Label 8033.2163
Modulus $8033$
Conductor $8033$
Order $276$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8033, base_ring=CyclotomicField(276))
 
M = H._module
 
chi = DirichletCharacter(H, M([207,205]))
 
pari: [g,chi] = znchar(Mod(2163,8033))
 

Basic properties

Modulus: \(8033\)
Conductor: \(8033\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(276\)
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 8033.ci

\(\chi_{8033}(99,\cdot)\) \(\chi_{8033}(447,\cdot)\) \(\chi_{8033}(476,\cdot)\) \(\chi_{8033}(568,\cdot)\) \(\chi_{8033}(626,\cdot)\) \(\chi_{8033}(650,\cdot)\) \(\chi_{8033}(655,\cdot)\) \(\chi_{8033}(737,\cdot)\) \(\chi_{8033}(800,\cdot)\) \(\chi_{8033}(974,\cdot)\) \(\chi_{8033}(1003,\cdot)\) \(\chi_{8033}(1090,\cdot)\) \(\chi_{8033}(1114,\cdot)\) \(\chi_{8033}(1201,\cdot)\) \(\chi_{8033}(1259,\cdot)\) \(\chi_{8033}(1288,\cdot)\) \(\chi_{8033}(1317,\cdot)\) \(\chi_{8033}(1409,\cdot)\) \(\chi_{8033}(1525,\cdot)\) \(\chi_{8033}(1786,\cdot)\) \(\chi_{8033}(1989,\cdot)\) \(\chi_{8033}(2042,\cdot)\) \(\chi_{8033}(2105,\cdot)\) \(\chi_{8033}(2158,\cdot)\) \(\chi_{8033}(2163,\cdot)\) \(\chi_{8033}(2221,\cdot)\) \(\chi_{8033}(2366,\cdot)\) \(\chi_{8033}(2395,\cdot)\) \(\chi_{8033}(2448,\cdot)\) \(\chi_{8033}(2482,\cdot)\) ...

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: $\Q(\zeta_{276})$
Fixed field: Number field defined by a degree 276 polynomial (not computed)

Values on generators

\((5541,1944)\) → \((-i,e\left(\frac{205}{276}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 8033 }(2163, a) \) \(1\)\(1\)\(e\left(\frac{43}{46}\right)\)\(e\left(\frac{107}{276}\right)\)\(e\left(\frac{20}{23}\right)\)\(e\left(\frac{67}{276}\right)\)\(e\left(\frac{89}{276}\right)\)\(e\left(\frac{47}{138}\right)\)\(e\left(\frac{37}{46}\right)\)\(e\left(\frac{107}{138}\right)\)\(e\left(\frac{49}{276}\right)\)\(e\left(\frac{131}{138}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8033 }(2163,a) \;\) at \(\;a = \) e.g. 2