Properties

Label 5824.53
Modulus $5824$
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(5824, base_ring=CyclotomicField(48))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,15,32,0]))
 
pari: [g,chi] = znchar(Mod(53,5824))
 

Basic properties

Modulus: \(5824\)
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 \(\chi_{448}(53,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5824.my

\(\chi_{5824}(53,\cdot)\) \(\chi_{5824}(261,\cdot)\) \(\chi_{5824}(781,\cdot)\) \(\chi_{5824}(989,\cdot)\) \(\chi_{5824}(1509,\cdot)\) \(\chi_{5824}(1717,\cdot)\) \(\chi_{5824}(2237,\cdot)\) \(\chi_{5824}(2445,\cdot)\) \(\chi_{5824}(2965,\cdot)\) \(\chi_{5824}(3173,\cdot)\) \(\chi_{5824}(3693,\cdot)\) \(\chi_{5824}(3901,\cdot)\) \(\chi_{5824}(4421,\cdot)\) \(\chi_{5824}(4629,\cdot)\) \(\chi_{5824}(5149,\cdot)\) \(\chi_{5824}(5357,\cdot)\)

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

\((2367,1093,4161,4929)\) → \((1,e\left(\frac{5}{16}\right),e\left(\frac{2}{3}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(15\)\(17\)\(19\)\(23\)\(25\)\(27\)
\( \chi_{ 5824 }(53, a) \) \(1\)\(1\)\(e\left(\frac{29}{48}\right)\)\(e\left(\frac{31}{48}\right)\)\(e\left(\frac{5}{24}\right)\)\(e\left(\frac{11}{48}\right)\)\(i\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{25}{48}\right)\)\(e\left(\frac{17}{24}\right)\)\(e\left(\frac{7}{24}\right)\)\(e\left(\frac{13}{16}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5824 }(53,a) \;\) at \(\;a = \) e.g. 2