Properties

Label 5824.251
Modulus $5824$
Conductor $5824$
Order $48$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5824, base_ring=CyclotomicField(48))
 
M = H._module
 
chi = DirichletCharacter(H, M([24,3,24,8]))
 
pari: [g,chi] = znchar(Mod(251,5824))
 

Basic properties

Modulus: \(5824\)
Conductor: \(5824\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(48\)
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 5824.ne

\(\chi_{5824}(251,\cdot)\) \(\chi_{5824}(699,\cdot)\) \(\chi_{5824}(979,\cdot)\) \(\chi_{5824}(1427,\cdot)\) \(\chi_{5824}(1707,\cdot)\) \(\chi_{5824}(2155,\cdot)\) \(\chi_{5824}(2435,\cdot)\) \(\chi_{5824}(2883,\cdot)\) \(\chi_{5824}(3163,\cdot)\) \(\chi_{5824}(3611,\cdot)\) \(\chi_{5824}(3891,\cdot)\) \(\chi_{5824}(4339,\cdot)\) \(\chi_{5824}(4619,\cdot)\) \(\chi_{5824}(5067,\cdot)\) \(\chi_{5824}(5347,\cdot)\) \(\chi_{5824}(5795,\cdot)\)

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

\((2367,1093,4161,4929)\) → \((-1,e\left(\frac{1}{16}\right),-1,e\left(\frac{1}{6}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(15\)\(17\)\(19\)\(23\)\(25\)\(27\)
\( \chi_{ 5824 }(251, a) \) \(1\)\(1\)\(e\left(\frac{41}{48}\right)\)\(e\left(\frac{1}{16}\right)\)\(e\left(\frac{17}{24}\right)\)\(e\left(\frac{47}{48}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{13}{48}\right)\)\(e\left(\frac{1}{24}\right)\)\(e\left(\frac{1}{8}\right)\)\(e\left(\frac{9}{16}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5824 }(251,a) \;\) at \(\;a = \) e.g. 2