Properties

Label 3024.565
Modulus $3024$
Conductor $3024$
Order $36$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3024, base_ring=CyclotomicField(36))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,9,20,30]))
 
pari: [g,chi] = znchar(Mod(565,3024))
 

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.hf

\(\chi_{3024}(61,\cdot)\) \(\chi_{3024}(157,\cdot)\) \(\chi_{3024}(565,\cdot)\) \(\chi_{3024}(661,\cdot)\) \(\chi_{3024}(1069,\cdot)\) \(\chi_{3024}(1165,\cdot)\) \(\chi_{3024}(1573,\cdot)\) \(\chi_{3024}(1669,\cdot)\) \(\chi_{3024}(2077,\cdot)\) \(\chi_{3024}(2173,\cdot)\) \(\chi_{3024}(2581,\cdot)\) \(\chi_{3024}(2677,\cdot)\)

sage: chi.galois_orbit()
 
order = charorder(g,chi)
 
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{36})\)
Fixed field: 36.0.13854191503209908618935635896029042342963612597539073353861947814992950774768801013176099229663232.2

Values on generators

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

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 3024 }(565, a) \) \(-1\)\(1\)\(e\left(\frac{7}{36}\right)\)\(e\left(\frac{29}{36}\right)\)\(e\left(\frac{25}{36}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{11}{36}\right)\)\(e\left(\frac{17}{18}\right)\)\(i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3024 }(565,a) \;\) at \(\;a = \) e.g. 2