Properties

Label 8009.273
Modulus $8009$
Conductor $8009$
Order $1144$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8009, base_ring=CyclotomicField(1144))
 
M = H._module
 
chi = DirichletCharacter(H, M([475]))
 
pari: [g,chi] = znchar(Mod(273,8009))
 

Basic properties

Modulus: \(8009\)
Conductor: \(8009\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(1144\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8009.bc

\(\chi_{8009}(15,\cdot)\) \(\chi_{8009}(21,\cdot)\) \(\chi_{8009}(24,\cdot)\) \(\chi_{8009}(76,\cdot)\) \(\chi_{8009}(87,\cdot)\) \(\chi_{8009}(94,\cdot)\) \(\chi_{8009}(108,\cdot)\) \(\chi_{8009}(109,\cdot)\) \(\chi_{8009}(131,\cdot)\) \(\chi_{8009}(132,\cdot)\) \(\chi_{8009}(151,\cdot)\) \(\chi_{8009}(158,\cdot)\) \(\chi_{8009}(194,\cdot)\) \(\chi_{8009}(195,\cdot)\) \(\chi_{8009}(197,\cdot)\) \(\chi_{8009}(205,\cdot)\) \(\chi_{8009}(258,\cdot)\) \(\chi_{8009}(273,\cdot)\) \(\chi_{8009}(287,\cdot)\) \(\chi_{8009}(312,\cdot)\) \(\chi_{8009}(379,\cdot)\) \(\chi_{8009}(418,\cdot)\) \(\chi_{8009}(423,\cdot)\) \(\chi_{8009}(426,\cdot)\) \(\chi_{8009}(449,\cdot)\) \(\chi_{8009}(458,\cdot)\) \(\chi_{8009}(459,\cdot)\) \(\chi_{8009}(486,\cdot)\) \(\chi_{8009}(487,\cdot)\) \(\chi_{8009}(489,\cdot)\) ...

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: $\Q(\zeta_{1144})$
Fixed field: Number field defined by a degree 1144 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{475}{1144}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 8009 }(273, a) \) \(-1\)\(1\)\(e\left(\frac{493}{572}\right)\)\(e\left(\frac{475}{1144}\right)\)\(e\left(\frac{207}{286}\right)\)\(e\left(\frac{17}{44}\right)\)\(e\left(\frac{317}{1144}\right)\)\(e\left(\frac{40}{143}\right)\)\(e\left(\frac{335}{572}\right)\)\(e\left(\frac{475}{572}\right)\)\(e\left(\frac{71}{286}\right)\)\(e\left(\frac{433}{572}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8009 }(273,a) \;\) at \(\;a = \) e.g. 2