Properties

Label 3024.1229
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([0,27,34,24]))
 
pari: [g,chi] = znchar(Mod(1229,3024))
 

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.gy

\(\chi_{3024}(221,\cdot)\) \(\chi_{3024}(317,\cdot)\) \(\chi_{3024}(725,\cdot)\) \(\chi_{3024}(821,\cdot)\) \(\chi_{3024}(1229,\cdot)\) \(\chi_{3024}(1325,\cdot)\) \(\chi_{3024}(1733,\cdot)\) \(\chi_{3024}(1829,\cdot)\) \(\chi_{3024}(2237,\cdot)\) \(\chi_{3024}(2333,\cdot)\) \(\chi_{3024}(2741,\cdot)\) \(\chi_{3024}(2837,\cdot)\)

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

\((1135,757,785,2593)\) → \((1,-i,e\left(\frac{17}{18}\right),e\left(\frac{2}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 3024 }(1229, a) \) \(-1\)\(1\)\(e\left(\frac{29}{36}\right)\)\(e\left(\frac{25}{36}\right)\)\(e\left(\frac{29}{36}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{7}{36}\right)\)\(e\left(\frac{5}{9}\right)\)\(-i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3024 }(1229,a) \;\) at \(\;a = \) e.g. 2