Properties

Label 3024.853
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([0,9,8,18]))
 
pari: [g,chi] = znchar(Mod(853,3024))
 

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.hd

\(\chi_{3024}(13,\cdot)\) \(\chi_{3024}(349,\cdot)\) \(\chi_{3024}(517,\cdot)\) \(\chi_{3024}(853,\cdot)\) \(\chi_{3024}(1021,\cdot)\) \(\chi_{3024}(1357,\cdot)\) \(\chi_{3024}(1525,\cdot)\) \(\chi_{3024}(1861,\cdot)\) \(\chi_{3024}(2029,\cdot)\) \(\chi_{3024}(2365,\cdot)\) \(\chi_{3024}(2533,\cdot)\) \(\chi_{3024}(2869,\cdot)\)

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: Number field defined by a degree 36 polynomial

Values on generators

\((1135,757,785,2593)\) → \((1,i,e\left(\frac{2}{9}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 3024 }(853, a) \) \(-1\)\(1\)\(e\left(\frac{31}{36}\right)\)\(e\left(\frac{5}{36}\right)\)\(e\left(\frac{1}{36}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{35}{36}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{7}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3024 }(853,a) \;\) at \(\;a = \) e.g. 2