Properties

Label 1344.859
Modulus $1344$
Conductor $448$
Order $48$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1344, base_ring=CyclotomicField(48))
 
M = H._module
 
chi = DirichletCharacter(H, M([24,27,0,40]))
 
pari: [g,chi] = znchar(Mod(859,1344))
 

Basic properties

Modulus: \(1344\)
Conductor: \(448\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(48\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{448}(411,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1344.cy

\(\chi_{1344}(19,\cdot)\) \(\chi_{1344}(115,\cdot)\) \(\chi_{1344}(187,\cdot)\) \(\chi_{1344}(283,\cdot)\) \(\chi_{1344}(355,\cdot)\) \(\chi_{1344}(451,\cdot)\) \(\chi_{1344}(523,\cdot)\) \(\chi_{1344}(619,\cdot)\) \(\chi_{1344}(691,\cdot)\) \(\chi_{1344}(787,\cdot)\) \(\chi_{1344}(859,\cdot)\) \(\chi_{1344}(955,\cdot)\) \(\chi_{1344}(1027,\cdot)\) \(\chi_{1344}(1123,\cdot)\) \(\chi_{1344}(1195,\cdot)\) \(\chi_{1344}(1291,\cdot)\)

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: \(\Q(\zeta_{48})\)
Fixed field: Number field defined by a degree 48 polynomial

Values on generators

\((127,1093,449,577)\) → \((-1,e\left(\frac{9}{16}\right),1,e\left(\frac{5}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 1344 }(859, a) \) \(1\)\(1\)\(e\left(\frac{35}{48}\right)\)\(e\left(\frac{31}{48}\right)\)\(e\left(\frac{15}{16}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{29}{48}\right)\)\(e\left(\frac{1}{24}\right)\)\(e\left(\frac{11}{24}\right)\)\(e\left(\frac{3}{16}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{35}{48}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1344 }(859,a) \;\) at \(\;a = \) e.g. 2