Properties

Label 4008.733
Modulus $4008$
Conductor $1336$
Order $166$
Real no
Primitive no
Minimal yes
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([0,83,0,104]))
 
pari: [g,chi] = znchar(Mod(733,4008))
 

Basic properties

Modulus: \(4008\)
Conductor: \(1336\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(166\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1336}(733,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4008.bb

\(\chi_{4008}(61,\cdot)\) \(\chi_{4008}(85,\cdot)\) \(\chi_{4008}(133,\cdot)\) \(\chi_{4008}(157,\cdot)\) \(\chi_{4008}(181,\cdot)\) \(\chi_{4008}(205,\cdot)\) \(\chi_{4008}(229,\cdot)\) \(\chi_{4008}(397,\cdot)\) \(\chi_{4008}(421,\cdot)\) \(\chi_{4008}(517,\cdot)\) \(\chi_{4008}(565,\cdot)\) \(\chi_{4008}(589,\cdot)\) \(\chi_{4008}(613,\cdot)\) \(\chi_{4008}(733,\cdot)\) \(\chi_{4008}(757,\cdot)\) \(\chi_{4008}(805,\cdot)\) \(\chi_{4008}(853,\cdot)\) \(\chi_{4008}(877,\cdot)\) \(\chi_{4008}(901,\cdot)\) \(\chi_{4008}(949,\cdot)\) \(\chi_{4008}(997,\cdot)\) \(\chi_{4008}(1021,\cdot)\) \(\chi_{4008}(1117,\cdot)\) \(\chi_{4008}(1213,\cdot)\) \(\chi_{4008}(1285,\cdot)\) \(\chi_{4008}(1357,\cdot)\) \(\chi_{4008}(1429,\cdot)\) \(\chi_{4008}(1477,\cdot)\) \(\chi_{4008}(1525,\cdot)\) \(\chi_{4008}(1597,\cdot)\) ...

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

\((3007,2005,1337,673)\) → \((1,-1,1,e\left(\frac{52}{83}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(1\)\(1\)\(e\left(\frac{21}{166}\right)\)\(e\left(\frac{77}{83}\right)\)\(e\left(\frac{7}{166}\right)\)\(e\left(\frac{5}{166}\right)\)\(e\left(\frac{17}{83}\right)\)\(e\left(\frac{139}{166}\right)\)\(e\left(\frac{2}{83}\right)\)\(e\left(\frac{21}{83}\right)\)\(e\left(\frac{79}{166}\right)\)\(e\left(\frac{32}{83}\right)\)
value at e.g. 2

Related number fields

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