Properties

Label 8004.239
Modulus $8004$
Conductor $8004$
Order $154$
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(154))
 
M = H._module
 
chi = DirichletCharacter(H, M([77,77,70,66]))
 
pari: [g,chi] = znchar(Mod(239,8004))
 

Basic properties

Modulus: \(8004\)
Conductor: \(8004\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(154\)
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.cy

\(\chi_{8004}(239,\cdot)\) \(\chi_{8004}(335,\cdot)\) \(\chi_{8004}(371,\cdot)\) \(\chi_{8004}(455,\cdot)\) \(\chi_{8004}(587,\cdot)\) \(\chi_{8004}(683,\cdot)\) \(\chi_{8004}(719,\cdot)\) \(\chi_{8004}(923,\cdot)\) \(\chi_{8004}(1067,\cdot)\) \(\chi_{8004}(1271,\cdot)\) \(\chi_{8004}(1283,\cdot)\) \(\chi_{8004}(1415,\cdot)\) \(\chi_{8004}(1475,\cdot)\) \(\chi_{8004}(1499,\cdot)\) \(\chi_{8004}(1619,\cdot)\) \(\chi_{8004}(1727,\cdot)\) \(\chi_{8004}(1823,\cdot)\) \(\chi_{8004}(1967,\cdot)\) \(\chi_{8004}(2111,\cdot)\) \(\chi_{8004}(2171,\cdot)\) \(\chi_{8004}(2327,\cdot)\) \(\chi_{8004}(2423,\cdot)\) \(\chi_{8004}(2519,\cdot)\) \(\chi_{8004}(2543,\cdot)\) \(\chi_{8004}(2663,\cdot)\) \(\chi_{8004}(2891,\cdot)\) \(\chi_{8004}(3155,\cdot)\) \(\chi_{8004}(3215,\cdot)\) \(\chi_{8004}(3371,\cdot)\) \(\chi_{8004}(3707,\cdot)\) ...

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: $\Q(\zeta_{77})$
Fixed field: Number field defined by a degree 154 polynomial (not computed)

Values on generators

\((4003,2669,3133,553)\) → \((-1,-1,e\left(\frac{5}{11}\right),e\left(\frac{3}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(25\)\(31\)\(35\)\(37\)
\( \chi_{ 8004 }(239, a) \) \(1\)\(1\)\(e\left(\frac{59}{154}\right)\)\(e\left(\frac{43}{154}\right)\)\(e\left(\frac{62}{77}\right)\)\(e\left(\frac{6}{77}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{27}{154}\right)\)\(e\left(\frac{59}{77}\right)\)\(e\left(\frac{101}{154}\right)\)\(e\left(\frac{51}{77}\right)\)\(e\left(\frac{64}{77}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8004 }(239,a) \;\) at \(\;a = \) e.g. 2