Properties

Label 5824.1097
Modulus $5824$
Conductor $2912$
Order $24$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5824, base_ring=CyclotomicField(24))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,9,20,18]))
 
pari: [g,chi] = znchar(Mod(1097,5824))
 

Basic properties

Modulus: \(5824\)
Conductor: \(2912\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(24\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2912}(733,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5824.ke

\(\chi_{5824}(1097,\cdot)\) \(\chi_{5824}(1529,\cdot)\) \(\chi_{5824}(2761,\cdot)\) \(\chi_{5824}(2777,\cdot)\) \(\chi_{5824}(4009,\cdot)\) \(\chi_{5824}(4441,\cdot)\) \(\chi_{5824}(5673,\cdot)\) \(\chi_{5824}(5689,\cdot)\)

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: \(\Q(\zeta_{24})\)
Fixed field: Number field defined by a degree 24 polynomial

Values on generators

\((2367,1093,4161,4929)\) → \((1,e\left(\frac{3}{8}\right),e\left(\frac{5}{6}\right),-i)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(15\)\(17\)\(19\)\(23\)\(25\)\(27\)
\( \chi_{ 5824 }(1097, a) \) \(1\)\(1\)\(e\left(\frac{23}{24}\right)\)\(e\left(\frac{7}{24}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{11}{24}\right)\)\(i\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{13}{24}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{7}{8}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5824 }(1097,a) \;\) at \(\;a = \) e.g. 2