Properties

Label 3024.2453
Modulus $3024$
Conductor $3024$
Order $36$
Real no
Primitive yes
Minimal yes
Parity even

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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([0,9,22,6]))
 
pari: [g,chi] = znchar(Mod(2453,3024))
 

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: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3024.gx

\(\chi_{3024}(173,\cdot)\) \(\chi_{3024}(437,\cdot)\) \(\chi_{3024}(677,\cdot)\) \(\chi_{3024}(941,\cdot)\) \(\chi_{3024}(1181,\cdot)\) \(\chi_{3024}(1445,\cdot)\) \(\chi_{3024}(1685,\cdot)\) \(\chi_{3024}(1949,\cdot)\) \(\chi_{3024}(2189,\cdot)\) \(\chi_{3024}(2453,\cdot)\) \(\chi_{3024}(2693,\cdot)\) \(\chi_{3024}(2957,\cdot)\)

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: 36.36.124687723528889177570420723064261381086672513377851660184757530334936556972919209118584893066969088.1

Values on generators

\((1135,757,785,2593)\) → \((1,i,e\left(\frac{11}{18}\right),e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 3024 }(2453, a) \) \(1\)\(1\)\(e\left(\frac{5}{36}\right)\)\(e\left(\frac{31}{36}\right)\)\(e\left(\frac{5}{36}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{13}{36}\right)\)\(e\left(\frac{7}{18}\right)\)\(i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3024 }(2453,a) \;\) at \(\;a = \) e.g. 2