Properties

Label 3024.397
Modulus $3024$
Conductor $1008$
Order $12$
Real no
Primitive no
Minimal no
Parity odd

Related objects

Learn more about

Show commands for: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(3024, base_ring=CyclotomicField(12))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,9,8,10]))
 
pari: [g,chi] = znchar(Mod(397,3024))
 

Basic properties

Modulus: \(3024\)
Conductor: \(1008\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1008}(61,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3024.ee

\(\chi_{3024}(397,\cdot)\) \(\chi_{3024}(1333,\cdot)\) \(\chi_{3024}(1909,\cdot)\) \(\chi_{3024}(2845,\cdot)\)

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

\((1135,757,785,2593)\) → \((1,-i,e\left(\frac{2}{3}\right),e\left(\frac{5}{6}\right))\)

Values

\(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\(-1\)\(1\)\(i\)\(-i\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{5}{12}\right)\)\(-1\)\(-1\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{5}{12}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{12})\)
Fixed field: 12.0.104450454135111752947335168.2