Properties

Label 4005.38
Modulus $4005$
Conductor $4005$
Order $264$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4005, base_ring=CyclotomicField(264))
 
M = H._module
 
chi = DirichletCharacter(H, M([44,198,153]))
 
pari: [g,chi] = znchar(Mod(38,4005))
 

Basic properties

Modulus: \(4005\)
Conductor: \(4005\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(264\)
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 4005.ee

\(\chi_{4005}(38,\cdot)\) \(\chi_{4005}(92,\cdot)\) \(\chi_{4005}(122,\cdot)\) \(\chi_{4005}(137,\cdot)\) \(\chi_{4005}(248,\cdot)\) \(\chi_{4005}(293,\cdot)\) \(\chi_{4005}(362,\cdot)\) \(\chi_{4005}(383,\cdot)\) \(\chi_{4005}(452,\cdot)\) \(\chi_{4005}(527,\cdot)\) \(\chi_{4005}(563,\cdot)\) \(\chi_{4005}(608,\cdot)\) \(\chi_{4005}(617,\cdot)\) \(\chi_{4005}(653,\cdot)\) \(\chi_{4005}(743,\cdot)\) \(\chi_{4005}(788,\cdot)\) \(\chi_{4005}(842,\cdot)\) \(\chi_{4005}(857,\cdot)\) \(\chi_{4005}(887,\cdot)\) \(\chi_{4005}(1022,\cdot)\) \(\chi_{4005}(1082,\cdot)\) \(\chi_{4005}(1103,\cdot)\) \(\chi_{4005}(1208,\cdot)\) \(\chi_{4005}(1292,\cdot)\) \(\chi_{4005}(1307,\cdot)\) \(\chi_{4005}(1373,\cdot)\) \(\chi_{4005}(1427,\cdot)\) \(\chi_{4005}(1472,\cdot)\) \(\chi_{4005}(1478,\cdot)\) \(\chi_{4005}(1667,\cdot)\) ...

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: $\Q(\zeta_{264})$
Fixed field: Number field defined by a degree 264 polynomial (not computed)

Values on generators

\((3116,802,181)\) → \((e\left(\frac{1}{6}\right),-i,e\left(\frac{51}{88}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 4005 }(38, a) \) \(-1\)\(1\)\(e\left(\frac{25}{132}\right)\)\(e\left(\frac{25}{66}\right)\)\(e\left(\frac{95}{264}\right)\)\(e\left(\frac{25}{44}\right)\)\(e\left(\frac{28}{33}\right)\)\(e\left(\frac{241}{264}\right)\)\(e\left(\frac{145}{264}\right)\)\(e\left(\frac{25}{33}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{69}{88}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4005 }(38,a) \;\) at \(\;a = \) e.g. 2