Properties

Label 3024.229
Modulus $3024$
Conductor $3024$
Order $36$
Real no
Primitive yes
Minimal yes
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(36))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,9,16,30]))
 
pari: [g,chi] = znchar(Mod(229,3024))
 

Basic properties

Modulus: \(3024\)
Conductor: \(3024\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(36\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3024.go

\(\chi_{3024}(229,\cdot)\) \(\chi_{3024}(493,\cdot)\) \(\chi_{3024}(733,\cdot)\) \(\chi_{3024}(997,\cdot)\) \(\chi_{3024}(1237,\cdot)\) \(\chi_{3024}(1501,\cdot)\) \(\chi_{3024}(1741,\cdot)\) \(\chi_{3024}(2005,\cdot)\) \(\chi_{3024}(2245,\cdot)\) \(\chi_{3024}(2509,\cdot)\) \(\chi_{3024}(2749,\cdot)\) \(\chi_{3024}(3013,\cdot)\)

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

\((1135,757,785,2593)\) → \((1,i,e\left(\frac{4}{9}\right),e\left(\frac{5}{6}\right))\)

Values

\(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\(-1\)\(1\)\(e\left(\frac{23}{36}\right)\)\(e\left(\frac{13}{36}\right)\)\(e\left(\frac{29}{36}\right)\)\(-1\)\(i\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{7}{36}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{7}{12}\right)\)
value at e.g. 2

Related number fields

Field of values: \(\Q(\zeta_{36})\)
Fixed field: 36.0.13854191503209908618935635896029042342963612597539073353861947814992950774768801013176099229663232.1