Properties

Label 4008.455
Modulus $4008$
Conductor $2004$
Order $166$
Real no
Primitive no
Minimal no
Parity even

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([83,0,83,56]))
 
pari: [g,chi] = znchar(Mod(455,4008))
 

Basic properties

Modulus: \(4008\)
Conductor: \(2004\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(166\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2004}(455,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4008.bc

\(\chi_{4008}(47,\cdot)\) \(\chi_{4008}(191,\cdot)\) \(\chi_{4008}(215,\cdot)\) \(\chi_{4008}(239,\cdot)\) \(\chi_{4008}(263,\cdot)\) \(\chi_{4008}(311,\cdot)\) \(\chi_{4008}(359,\cdot)\) \(\chi_{4008}(383,\cdot)\) \(\chi_{4008}(431,\cdot)\) \(\chi_{4008}(455,\cdot)\) \(\chi_{4008}(503,\cdot)\) \(\chi_{4008}(551,\cdot)\) \(\chi_{4008}(599,\cdot)\) \(\chi_{4008}(623,\cdot)\) \(\chi_{4008}(671,\cdot)\) \(\chi_{4008}(695,\cdot)\) \(\chi_{4008}(743,\cdot)\) \(\chi_{4008}(767,\cdot)\) \(\chi_{4008}(815,\cdot)\) \(\chi_{4008}(839,\cdot)\) \(\chi_{4008}(863,\cdot)\) \(\chi_{4008}(911,\cdot)\) \(\chi_{4008}(935,\cdot)\) \(\chi_{4008}(959,\cdot)\) \(\chi_{4008}(1031,\cdot)\) \(\chi_{4008}(1079,\cdot)\) \(\chi_{4008}(1175,\cdot)\) \(\chi_{4008}(1223,\cdot)\) \(\chi_{4008}(1295,\cdot)\) \(\chi_{4008}(1319,\cdot)\) ...

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

Values on generators

\((3007,2005,1337,673)\) → \((-1,1,-1,e\left(\frac{28}{83}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(1\)\(1\)\(e\left(\frac{139}{166}\right)\)\(e\left(\frac{51}{166}\right)\)\(e\left(\frac{37}{83}\right)\)\(e\left(\frac{62}{83}\right)\)\(e\left(\frac{63}{166}\right)\)\(e\left(\frac{11}{166}\right)\)\(e\left(\frac{33}{83}\right)\)\(e\left(\frac{56}{83}\right)\)\(e\left(\frac{17}{166}\right)\)\(e\left(\frac{143}{166}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{83})$
Fixed field: Number field defined by a degree 166 polynomial