Properties

Label 8033.644
Modulus $8033$
Conductor $8033$
Order $966$
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(966))
 
M = H._module
 
chi = DirichletCharacter(H, M([207,868]))
 
pari: [g,chi] = znchar(Mod(644,8033))
 

Basic properties

Modulus: \(8033\)
Conductor: \(8033\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(966\)
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.cu

\(\chi_{8033}(9,\cdot)\) \(\chi_{8033}(67,\cdot)\) \(\chi_{8033}(71,\cdot)\) \(\chi_{8033}(91,\cdot)\) \(\chi_{8033}(100,\cdot)\) \(\chi_{8033}(154,\cdot)\) \(\chi_{8033}(207,\cdot)\) \(\chi_{8033}(237,\cdot)\) \(\chi_{8033}(238,\cdot)\) \(\chi_{8033}(241,\cdot)\) \(\chi_{8033}(265,\cdot)\) \(\chi_{8033}(270,\cdot)\) \(\chi_{8033}(325,\cdot)\) \(\chi_{8033}(332,\cdot)\) \(\chi_{8033}(448,\cdot)\) \(\chi_{8033}(468,\cdot)\) \(\chi_{8033}(515,\cdot)\) \(\chi_{8033}(557,\cdot)\) \(\chi_{8033}(564,\cdot)\) \(\chi_{8033}(602,\cdot)\) \(\chi_{8033}(642,\cdot)\) \(\chi_{8033}(644,\cdot)\) \(\chi_{8033}(701,\cdot)\) \(\chi_{8033}(792,\cdot)\) \(\chi_{8033}(834,\cdot)\) \(\chi_{8033}(854,\cdot)\) \(\chi_{8033}(879,\cdot)\) \(\chi_{8033}(912,\cdot)\) \(\chi_{8033}(921,\cdot)\) \(\chi_{8033}(1019,\cdot)\) ...

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: $\Q(\zeta_{483})$
Fixed field: Number field defined by a degree 966 polynomial (not computed)

Values on generators

\((5541,1944)\) → \((e\left(\frac{3}{14}\right),e\left(\frac{62}{69}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 8033 }(644, a) \) \(1\)\(1\)\(e\left(\frac{97}{322}\right)\)\(e\left(\frac{965}{966}\right)\)\(e\left(\frac{97}{161}\right)\)\(e\left(\frac{296}{483}\right)\)\(e\left(\frac{145}{483}\right)\)\(e\left(\frac{164}{483}\right)\)\(e\left(\frac{291}{322}\right)\)\(e\left(\frac{482}{483}\right)\)\(e\left(\frac{883}{966}\right)\)\(e\left(\frac{625}{966}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8033 }(644,a) \;\) at \(\;a = \) e.g. 2