Properties

Label 6003.289
Modulus $6003$
Conductor $667$
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([0,14,11]))
 
pari: [g,chi] = znchar(Mod(289,6003))
 

Basic properties

Modulus: \(6003\)
Conductor: \(667\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{667}(289,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6003.bl

\(\chi_{6003}(289,\cdot)\) \(\chi_{6003}(811,\cdot)\) \(\chi_{6003}(2377,\cdot)\) \(\chi_{6003}(2638,\cdot)\) \(\chi_{6003}(3160,\cdot)\) \(\chi_{6003}(3682,\cdot)\) \(\chi_{6003}(4204,\cdot)\) \(\chi_{6003}(4465,\cdot)\) \(\chi_{6003}(5248,\cdot)\) \(\chi_{6003}(5509,\cdot)\)

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

\((668,3133,4555)\) → \((1,e\left(\frac{7}{11}\right),-1)\)

Values

\(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\(1\)\(1\)\(e\left(\frac{17}{22}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{1}{11}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{11})\)
Fixed field: 22.22.20937975979670626213353681795476767790826629.1