Properties

Label 6048.1315
Modulus $6048$
Conductor $2016$
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(6048, base_ring=CyclotomicField(24))
 
M = H._module
 
chi = DirichletCharacter(H, M([12,9,16,12]))
 
pari: [g,chi] = znchar(Mod(1315,6048))
 

Basic properties

Modulus: \(6048\)
Conductor: \(2016\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(24\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2016}(1987,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6048.hx

\(\chi_{6048}(307,\cdot)\) \(\chi_{6048}(1315,\cdot)\) \(\chi_{6048}(1819,\cdot)\) \(\chi_{6048}(2827,\cdot)\) \(\chi_{6048}(3331,\cdot)\) \(\chi_{6048}(4339,\cdot)\) \(\chi_{6048}(4843,\cdot)\) \(\chi_{6048}(5851,\cdot)\)

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

\((4159,3781,3809,2593)\) → \((-1,e\left(\frac{3}{8}\right),e\left(\frac{2}{3}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 6048 }(1315, a) \) \(1\)\(1\)\(e\left(\frac{5}{24}\right)\)\(e\left(\frac{1}{24}\right)\)\(e\left(\frac{11}{24}\right)\)\(1\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{19}{24}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{3}{8}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6048 }(1315,a) \;\) at \(\;a = \) e.g. 2