Properties

Label 4008.269
Modulus $4008$
Conductor $4008$
Order $166$
Real no
Primitive yes
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([0,83,83,21]))
 
pari: [g,chi] = znchar(Mod(269,4008))
 

Basic properties

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

Galois orbit 4008.y

\(\chi_{4008}(5,\cdot)\) \(\chi_{4008}(53,\cdot)\) \(\chi_{4008}(101,\cdot)\) \(\chi_{4008}(125,\cdot)\) \(\chi_{4008}(149,\cdot)\) \(\chi_{4008}(197,\cdot)\) \(\chi_{4008}(245,\cdot)\) \(\chi_{4008}(269,\cdot)\) \(\chi_{4008}(389,\cdot)\) \(\chi_{4008}(413,\cdot)\) \(\chi_{4008}(437,\cdot)\) \(\chi_{4008}(485,\cdot)\) \(\chi_{4008}(581,\cdot)\) \(\chi_{4008}(605,\cdot)\) \(\chi_{4008}(773,\cdot)\) \(\chi_{4008}(797,\cdot)\) \(\chi_{4008}(821,\cdot)\) \(\chi_{4008}(845,\cdot)\) \(\chi_{4008}(869,\cdot)\) \(\chi_{4008}(917,\cdot)\) \(\chi_{4008}(941,\cdot)\) \(\chi_{4008}(1037,\cdot)\) \(\chi_{4008}(1061,\cdot)\) \(\chi_{4008}(1085,\cdot)\) \(\chi_{4008}(1133,\cdot)\) \(\chi_{4008}(1157,\cdot)\) \(\chi_{4008}(1229,\cdot)\) \(\chi_{4008}(1325,\cdot)\) \(\chi_{4008}(1349,\cdot)\) \(\chi_{4008}(1373,\cdot)\) ...

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

\((3007,2005,1337,673)\) → \((1,-1,-1,e\left(\frac{21}{166}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(1\)\(1\)\(e\left(\frac{21}{166}\right)\)\(e\left(\frac{77}{83}\right)\)\(e\left(\frac{45}{83}\right)\)\(e\left(\frac{44}{83}\right)\)\(e\left(\frac{17}{83}\right)\)\(e\left(\frac{139}{166}\right)\)\(e\left(\frac{2}{83}\right)\)\(e\left(\frac{21}{83}\right)\)\(e\left(\frac{81}{83}\right)\)\(e\left(\frac{32}{83}\right)\)
value at e.g. 2

Related number fields

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