Properties

Label 6003.20
Modulus $6003$
Conductor $6003$
Order $462$
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([77,105,396]))
 
pari: [g,chi] = znchar(Mod(20,6003))
 

Basic properties

Modulus: \(6003\)
Conductor: \(6003\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(462\)
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.dl

\(\chi_{6003}(20,\cdot)\) \(\chi_{6003}(65,\cdot)\) \(\chi_{6003}(74,\cdot)\) \(\chi_{6003}(83,\cdot)\) \(\chi_{6003}(194,\cdot)\) \(\chi_{6003}(227,\cdot)\) \(\chi_{6003}(281,\cdot)\) \(\chi_{6003}(401,\cdot)\) \(\chi_{6003}(488,\cdot)\) \(\chi_{6003}(500,\cdot)\) \(\chi_{6003}(596,\cdot)\) \(\chi_{6003}(605,\cdot)\) \(\chi_{6003}(632,\cdot)\) \(\chi_{6003}(770,\cdot)\) \(\chi_{6003}(779,\cdot)\) \(\chi_{6003}(803,\cdot)\) \(\chi_{6003}(848,\cdot)\) \(\chi_{6003}(866,\cdot)\) \(\chi_{6003}(893,\cdot)\) \(\chi_{6003}(977,\cdot)\) \(\chi_{6003}(1010,\cdot)\) \(\chi_{6003}(1022,\cdot)\) \(\chi_{6003}(1031,\cdot)\) \(\chi_{6003}(1040,\cdot)\) \(\chi_{6003}(1109,\cdot)\) \(\chi_{6003}(1118,\cdot)\) \(\chi_{6003}(1184,\cdot)\) \(\chi_{6003}(1238,\cdot)\) \(\chi_{6003}(1325,\cdot)\) \(\chi_{6003}(1328,\cdot)\) ...

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

\((668,3133,4555)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{5}{22}\right),e\left(\frac{6}{7}\right))\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(1\)\(1\)\(e\left(\frac{221}{462}\right)\)\(e\left(\frac{221}{231}\right)\)\(e\left(\frac{212}{231}\right)\)\(e\left(\frac{125}{462}\right)\)\(e\left(\frac{67}{154}\right)\)\(e\left(\frac{61}{154}\right)\)\(e\left(\frac{148}{231}\right)\)\(e\left(\frac{218}{231}\right)\)\(e\left(\frac{173}{231}\right)\)\(e\left(\frac{211}{231}\right)\)
value at e.g. 2

Related number fields

Field of values: $\Q(\zeta_{231})$
Fixed field: Number field defined by a degree 462 polynomial