Properties

Label 4002.85
Modulus $4002$
Conductor $667$
Order $308$
Real no
Primitive no
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4002, base_ring=CyclotomicField(308))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,112,165]))
 
pari: [g,chi] = znchar(Mod(85,4002))
 

Basic properties

Modulus: \(4002\)
Conductor: \(667\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(308\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{667}(85,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4002.bt

\(\chi_{4002}(31,\cdot)\) \(\chi_{4002}(55,\cdot)\) \(\chi_{4002}(73,\cdot)\) \(\chi_{4002}(85,\cdot)\) \(\chi_{4002}(127,\cdot)\) \(\chi_{4002}(163,\cdot)\) \(\chi_{4002}(193,\cdot)\) \(\chi_{4002}(211,\cdot)\) \(\chi_{4002}(259,\cdot)\) \(\chi_{4002}(271,\cdot)\) \(\chi_{4002}(301,\cdot)\) \(\chi_{4002}(403,\cdot)\) \(\chi_{4002}(409,\cdot)\) \(\chi_{4002}(427,\cdot)\) \(\chi_{4002}(445,\cdot)\) \(\chi_{4002}(541,\cdot)\) \(\chi_{4002}(565,\cdot)\) \(\chi_{4002}(577,\cdot)\) \(\chi_{4002}(583,\cdot)\) \(\chi_{4002}(601,\cdot)\) \(\chi_{4002}(607,\cdot)\) \(\chi_{4002}(685,\cdot)\) \(\chi_{4002}(715,\cdot)\) \(\chi_{4002}(739,\cdot)\) \(\chi_{4002}(775,\cdot)\) \(\chi_{4002}(823,\cdot)\) \(\chi_{4002}(859,\cdot)\) \(\chi_{4002}(901,\cdot)\) \(\chi_{4002}(913,\cdot)\) \(\chi_{4002}(949,\cdot)\) ...

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

\((2669,3133,553)\) → \((1,e\left(\frac{4}{11}\right),e\left(\frac{15}{28}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(25\)\(31\)\(35\)\(37\)
\( \chi_{ 4002 }(85, a) \) \(-1\)\(1\)\(e\left(\frac{23}{154}\right)\)\(e\left(\frac{26}{77}\right)\)\(e\left(\frac{205}{308}\right)\)\(e\left(\frac{113}{154}\right)\)\(e\left(\frac{35}{44}\right)\)\(e\left(\frac{85}{308}\right)\)\(e\left(\frac{23}{77}\right)\)\(e\left(\frac{221}{308}\right)\)\(e\left(\frac{75}{154}\right)\)\(e\left(\frac{75}{308}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4002 }(85,a) \;\) at \(\;a = \) e.g. 2