Properties

Label 6003.157
Modulus $6003$
Conductor $6003$
Order $132$
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(6003)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([44,90,33]))
 
pari: [g,chi] = znchar(Mod(157,6003))
 

Basic properties

Modulus: \(6003\)
Conductor: \(6003\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(132\)
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 6003.ct

\(\chi_{6003}(157,\cdot)\) \(\chi_{6003}(481,\cdot)\) \(\chi_{6003}(592,\cdot)\) \(\chi_{6003}(655,\cdot)\) \(\chi_{6003}(916,\cdot)\) \(\chi_{6003}(940,\cdot)\) \(\chi_{6003}(1003,\cdot)\) \(\chi_{6003}(1114,\cdot)\) \(\chi_{6003}(1201,\cdot)\) \(\chi_{6003}(1525,\cdot)\) \(\chi_{6003}(1699,\cdot)\) \(\chi_{6003}(1723,\cdot)\) \(\chi_{6003}(1786,\cdot)\) \(\chi_{6003}(1897,\cdot)\) \(\chi_{6003}(1960,\cdot)\) \(\chi_{6003}(2158,\cdot)\) \(\chi_{6003}(2245,\cdot)\) \(\chi_{6003}(2482,\cdot)\) \(\chi_{6003}(2767,\cdot)\) \(\chi_{6003}(2941,\cdot)\) \(\chi_{6003}(3004,\cdot)\) \(\chi_{6003}(3028,\cdot)\) \(\chi_{6003}(3202,\cdot)\) \(\chi_{6003}(3352,\cdot)\) \(\chi_{6003}(3526,\cdot)\) \(\chi_{6003}(3724,\cdot)\) \(\chi_{6003}(3787,\cdot)\) \(\chi_{6003}(3874,\cdot)\) \(\chi_{6003}(4246,\cdot)\) \(\chi_{6003}(4594,\cdot)\) ...

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

\((668,3133,4555)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{15}{22}\right),i)\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(1\)\(1\)\(e\left(\frac{125}{132}\right)\)\(e\left(\frac{59}{66}\right)\)\(e\left(\frac{28}{33}\right)\)\(e\left(\frac{19}{66}\right)\)\(e\left(\frac{37}{44}\right)\)\(e\left(\frac{35}{44}\right)\)\(e\left(\frac{95}{132}\right)\)\(e\left(\frac{47}{66}\right)\)\(e\left(\frac{31}{132}\right)\)\(e\left(\frac{26}{33}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{132})$
Fixed field: Number field defined by a degree 132 polynomial