Properties

Label 4008.407
Modulus $4008$
Conductor $2004$
Order $166$
Real no
Primitive no
Minimal no
Parity odd

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([83,0,83,89]))
 
pari: [g,chi] = znchar(Mod(407,4008))
 

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 \(\chi_{2004}(407,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4008.v

\(\chi_{4008}(23,\cdot)\) \(\chi_{4008}(71,\cdot)\) \(\chi_{4008}(95,\cdot)\) \(\chi_{4008}(119,\cdot)\) \(\chi_{4008}(143,\cdot)\) \(\chi_{4008}(287,\cdot)\) \(\chi_{4008}(407,\cdot)\) \(\chi_{4008}(479,\cdot)\) \(\chi_{4008}(527,\cdot)\) \(\chi_{4008}(575,\cdot)\) \(\chi_{4008}(647,\cdot)\) \(\chi_{4008}(719,\cdot)\) \(\chi_{4008}(791,\cdot)\) \(\chi_{4008}(887,\cdot)\) \(\chi_{4008}(983,\cdot)\) \(\chi_{4008}(1007,\cdot)\) \(\chi_{4008}(1055,\cdot)\) \(\chi_{4008}(1103,\cdot)\) \(\chi_{4008}(1127,\cdot)\) \(\chi_{4008}(1151,\cdot)\) \(\chi_{4008}(1199,\cdot)\) \(\chi_{4008}(1247,\cdot)\) \(\chi_{4008}(1271,\cdot)\) \(\chi_{4008}(1391,\cdot)\) \(\chi_{4008}(1415,\cdot)\) \(\chi_{4008}(1439,\cdot)\) \(\chi_{4008}(1487,\cdot)\) \(\chi_{4008}(1583,\cdot)\) \(\chi_{4008}(1607,\cdot)\) \(\chi_{4008}(1775,\cdot)\) ...

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

\((3007,2005,1337,673)\) → \((-1,1,-1,e\left(\frac{89}{166}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(-1\)\(1\)\(e\left(\frac{3}{83}\right)\)\(e\left(\frac{127}{166}\right)\)\(e\left(\frac{1}{83}\right)\)\(e\left(\frac{37}{166}\right)\)\(e\left(\frac{76}{83}\right)\)\(e\left(\frac{99}{166}\right)\)\(e\left(\frac{13}{166}\right)\)\(e\left(\frac{6}{83}\right)\)\(e\left(\frac{153}{166}\right)\)\(e\left(\frac{125}{166}\right)\)
value at e.g. 2

Related number fields

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