Properties

Label 6003.494
Modulus $6003$
Conductor $69$
Order $22$
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(6003)
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([11,9,0]))
 
pari: [g,chi] = znchar(Mod(494,6003))
 

Basic properties

Modulus: \(6003\)
Conductor: \(69\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{69}(11,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6003.bm

\(\chi_{6003}(494,\cdot)\) \(\chi_{6003}(755,\cdot)\) \(\chi_{6003}(1538,\cdot)\) \(\chi_{6003}(1799,\cdot)\) \(\chi_{6003}(2321,\cdot)\) \(\chi_{6003}(2843,\cdot)\) \(\chi_{6003}(3365,\cdot)\) \(\chi_{6003}(3626,\cdot)\) \(\chi_{6003}(5192,\cdot)\) \(\chi_{6003}(5714,\cdot)\)

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

\((668,3133,4555)\) → \((-1,e\left(\frac{9}{22}\right),1)\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(1\)\(1\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{3}{11}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{11})\)
Fixed field: \(\Q(\zeta_{69})^+\)