Properties

Label 8004.11
Modulus $8004$
Conductor $8004$
Order $308$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8004, base_ring=CyclotomicField(308))
 
M = H._module
 
chi = DirichletCharacter(H, M([154,154,126,275]))
 
pari: [g,chi] = znchar(Mod(11,8004))
 

Basic properties

Modulus: \(8004\)
Conductor: \(8004\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(308\)
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 8004.dm

\(\chi_{8004}(11,\cdot)\) \(\chi_{8004}(143,\cdot)\) \(\chi_{8004}(155,\cdot)\) \(\chi_{8004}(251,\cdot)\) \(\chi_{8004}(263,\cdot)\) \(\chi_{8004}(287,\cdot)\) \(\chi_{8004}(359,\cdot)\) \(\chi_{8004}(467,\cdot)\) \(\chi_{8004}(479,\cdot)\) \(\chi_{8004}(503,\cdot)\) \(\chi_{8004}(635,\cdot)\) \(\chi_{8004}(659,\cdot)\) \(\chi_{8004}(707,\cdot)\) \(\chi_{8004}(743,\cdot)\) \(\chi_{8004}(815,\cdot)\) \(\chi_{8004}(839,\cdot)\) \(\chi_{8004}(971,\cdot)\) \(\chi_{8004}(983,\cdot)\) \(\chi_{8004}(1055,\cdot)\) \(\chi_{8004}(1091,\cdot)\) \(\chi_{8004}(1187,\cdot)\) \(\chi_{8004}(1295,\cdot)\) \(\chi_{8004}(1307,\cdot)\) \(\chi_{8004}(1331,\cdot)\) \(\chi_{8004}(1355,\cdot)\) \(\chi_{8004}(1487,\cdot)\) \(\chi_{8004}(1523,\cdot)\) \(\chi_{8004}(1535,\cdot)\) \(\chi_{8004}(1643,\cdot)\) \(\chi_{8004}(1667,\cdot)\) ...

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: $\Q(\zeta_{308})$
Fixed field: Number field defined by a degree 308 polynomial (not computed)

Values on generators

\((4003,2669,3133,553)\) → \((-1,-1,e\left(\frac{9}{22}\right),e\left(\frac{25}{28}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(25\)\(31\)\(35\)\(37\)
\( \chi_{ 8004 }(11, a) \) \(1\)\(1\)\(e\left(\frac{85}{154}\right)\)\(e\left(\frac{76}{77}\right)\)\(e\left(\frac{1}{308}\right)\)\(e\left(\frac{123}{154}\right)\)\(e\left(\frac{5}{44}\right)\)\(e\left(\frac{207}{308}\right)\)\(e\left(\frac{8}{77}\right)\)\(e\left(\frac{261}{308}\right)\)\(e\left(\frac{83}{154}\right)\)\(e\left(\frac{83}{308}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8004 }(11,a) \;\) at \(\;a = \) e.g. 2