Properties

Label 8004.31
Modulus $8004$
Conductor $2668$
Order $308$
Real no
Primitive no
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([154,0,84,11]))
 
pari: [g,chi] = znchar(Mod(31,8004))
 

Basic properties

Modulus: \(8004\)
Conductor: \(2668\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(308\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2668}(31,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8004.dk

\(\chi_{8004}(31,\cdot)\) \(\chi_{8004}(55,\cdot)\) \(\chi_{8004}(127,\cdot)\) \(\chi_{8004}(163,\cdot)\) \(\chi_{8004}(211,\cdot)\) \(\chi_{8004}(259,\cdot)\) \(\chi_{8004}(271,\cdot)\) \(\chi_{8004}(403,\cdot)\) \(\chi_{8004}(427,\cdot)\) \(\chi_{8004}(583,\cdot)\) \(\chi_{8004}(607,\cdot)\) \(\chi_{8004}(715,\cdot)\) \(\chi_{8004}(739,\cdot)\) \(\chi_{8004}(775,\cdot)\) \(\chi_{8004}(823,\cdot)\) \(\chi_{8004}(859,\cdot)\) \(\chi_{8004}(955,\cdot)\) \(\chi_{8004}(1087,\cdot)\) \(\chi_{8004}(1099,\cdot)\) \(\chi_{8004}(1255,\cdot)\) \(\chi_{8004}(1291,\cdot)\) \(\chi_{8004}(1315,\cdot)\) \(\chi_{8004}(1411,\cdot)\) \(\chi_{8004}(1435,\cdot)\) \(\chi_{8004}(1603,\cdot)\) \(\chi_{8004}(1639,\cdot)\) \(\chi_{8004}(1651,\cdot)\) \(\chi_{8004}(1783,\cdot)\) \(\chi_{8004}(1819,\cdot)\) \(\chi_{8004}(1867,\cdot)\) ...

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

\((4003,2669,3133,553)\) → \((-1,1,e\left(\frac{3}{11}\right),e\left(\frac{1}{28}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(25\)\(31\)\(35\)\(37\)
\( \chi_{ 8004 }(31, a) \) \(1\)\(1\)\(e\left(\frac{9}{154}\right)\)\(e\left(\frac{17}{154}\right)\)\(e\left(\frac{261}{308}\right)\)\(e\left(\frac{71}{154}\right)\)\(e\left(\frac{29}{44}\right)\)\(e\left(\frac{281}{308}\right)\)\(e\left(\frac{9}{77}\right)\)\(e\left(\frac{53}{308}\right)\)\(e\left(\frac{13}{77}\right)\)\(e\left(\frac{257}{308}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8004 }(31,a) \;\) at \(\;a = \) e.g. 2