Properties

Label 4008.1115
Modulus $4008$
Conductor $4008$
Order $166$
Real no
Primitive yes
Minimal yes
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([83,83,83,73]))
 
pari: [g,chi] = znchar(Mod(1115,4008))
 

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: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4008.be

\(\chi_{4008}(35,\cdot)\) \(\chi_{4008}(59,\cdot)\) \(\chi_{4008}(83,\cdot)\) \(\chi_{4008}(131,\cdot)\) \(\chi_{4008}(155,\cdot)\) \(\chi_{4008}(227,\cdot)\) \(\chi_{4008}(323,\cdot)\) \(\chi_{4008}(347,\cdot)\) \(\chi_{4008}(371,\cdot)\) \(\chi_{4008}(443,\cdot)\) \(\chi_{4008}(587,\cdot)\) \(\chi_{4008}(611,\cdot)\) \(\chi_{4008}(635,\cdot)\) \(\chi_{4008}(659,\cdot)\) \(\chi_{4008}(683,\cdot)\) \(\chi_{4008}(707,\cdot)\) \(\chi_{4008}(779,\cdot)\) \(\chi_{4008}(803,\cdot)\) \(\chi_{4008}(827,\cdot)\) \(\chi_{4008}(875,\cdot)\) \(\chi_{4008}(971,\cdot)\) \(\chi_{4008}(995,\cdot)\) \(\chi_{4008}(1019,\cdot)\) \(\chi_{4008}(1043,\cdot)\) \(\chi_{4008}(1115,\cdot)\) \(\chi_{4008}(1163,\cdot)\) \(\chi_{4008}(1259,\cdot)\) \(\chi_{4008}(1307,\cdot)\) \(\chi_{4008}(1379,\cdot)\) \(\chi_{4008}(1403,\cdot)\) ...

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

\((3007,2005,1337,673)\) → \((-1,-1,-1,e\left(\frac{73}{166}\right))\)

Values

\(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\(-1\)\(1\)\(e\left(\frac{73}{166}\right)\)\(e\left(\frac{65}{166}\right)\)\(e\left(\frac{135}{166}\right)\)\(e\left(\frac{66}{83}\right)\)\(e\left(\frac{67}{83}\right)\)\(e\left(\frac{42}{83}\right)\)\(e\left(\frac{89}{166}\right)\)\(e\left(\frac{73}{83}\right)\)\(e\left(\frac{80}{83}\right)\)\(e\left(\frac{13}{166}\right)\)
value at e.g. 2

Related number fields

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