Properties

Label 6384.53
Modulus $6384$
Conductor $6384$
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(6384, base_ring=CyclotomicField(36))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,9,18,24,22]))
 
pari: [g,chi] = znchar(Mod(53,6384))
 

Basic properties

Modulus: \(6384\)
Conductor: \(6384\)
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 6384.nw

\(\chi_{6384}(53,\cdot)\) \(\chi_{6384}(1229,\cdot)\) \(\chi_{6384}(1325,\cdot)\) \(\chi_{6384}(1397,\cdot)\) \(\chi_{6384}(1997,\cdot)\) \(\chi_{6384}(3005,\cdot)\) \(\chi_{6384}(3245,\cdot)\) \(\chi_{6384}(4421,\cdot)\) \(\chi_{6384}(4517,\cdot)\) \(\chi_{6384}(4589,\cdot)\) \(\chi_{6384}(5189,\cdot)\) \(\chi_{6384}(6197,\cdot)\)

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

\((799,4789,2129,913,1009)\) → \((1,i,-1,e\left(\frac{2}{3}\right),e\left(\frac{11}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(23\)\(25\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 6384 }(53, a) \) \(1\)\(1\)\(e\left(\frac{31}{36}\right)\)\(-i\)\(e\left(\frac{29}{36}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{23}{36}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{17}{18}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6384 }(53,a) \;\) at \(\;a = \) e.g. 2