Properties

Modulus 4008
Conductor 1336
Order 166
Real no
Primitive no
Minimal yes
Parity even
Orbit label 4008.u

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(4008)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([83,83,0,165]))
 
pari: [g,chi] = znchar(Mod(67,4008))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Modulus = 4008
Conductor = 1336
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 166
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = no
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = even
Orbit label = 4008.u
Orbit index = 21

Galois orbit

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

\(\chi_{4008}(43,\cdot)\) \(\chi_{4008}(67,\cdot)\) \(\chi_{4008}(91,\cdot)\) \(\chi_{4008}(139,\cdot)\) \(\chi_{4008}(163,\cdot)\) \(\chi_{4008}(187,\cdot)\) \(\chi_{4008}(235,\cdot)\) \(\chi_{4008}(259,\cdot)\) \(\chi_{4008}(307,\cdot)\) \(\chi_{4008}(331,\cdot)\) \(\chi_{4008}(379,\cdot)\) \(\chi_{4008}(403,\cdot)\) \(\chi_{4008}(451,\cdot)\) \(\chi_{4008}(499,\cdot)\) \(\chi_{4008}(547,\cdot)\) \(\chi_{4008}(571,\cdot)\) \(\chi_{4008}(619,\cdot)\) \(\chi_{4008}(643,\cdot)\) \(\chi_{4008}(691,\cdot)\) \(\chi_{4008}(739,\cdot)\) \(\chi_{4008}(763,\cdot)\) \(\chi_{4008}(787,\cdot)\) \(\chi_{4008}(811,\cdot)\) \(\chi_{4008}(955,\cdot)\) \(\chi_{4008}(1075,\cdot)\) \(\chi_{4008}(1147,\cdot)\) \(\chi_{4008}(1195,\cdot)\) \(\chi_{4008}(1243,\cdot)\) \(\chi_{4008}(1315,\cdot)\) \(\chi_{4008}(1387,\cdot)\) ...

Values on generators

\((3007,2005,1337,673)\) → \((-1,-1,1,e\left(\frac{165}{166}\right))\)

Values

-11571113171923252931
\(1\)\(1\)\(e\left(\frac{41}{83}\right)\)\(e\left(\frac{131}{166}\right)\)\(e\left(\frac{69}{83}\right)\)\(e\left(\frac{73}{83}\right)\)\(e\left(\frac{113}{166}\right)\)\(e\left(\frac{54}{83}\right)\)\(e\left(\frac{75}{83}\right)\)\(e\left(\frac{82}{83}\right)\)\(e\left(\frac{99}{166}\right)\)\(e\left(\frac{159}{166}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{83})\)