Properties

Label 8007.292
Modulus $8007$
Conductor $2669$
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(8007, base_ring=CyclotomicField(48))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,3,28]))
 
pari: [g,chi] = znchar(Mod(292,8007))
 

Basic properties

Modulus: \(8007\)
Conductor: \(2669\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(48\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2669}(292,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8007.cv

\(\chi_{8007}(292,\cdot)\) \(\chi_{8007}(1306,\cdot)\) \(\chi_{8007}(1435,\cdot)\) \(\chi_{8007}(1705,\cdot)\) \(\chi_{8007}(2305,\cdot)\) \(\chi_{8007}(2647,\cdot)\) \(\chi_{8007}(4060,\cdot)\) \(\chi_{8007}(4189,\cdot)\) \(\chi_{8007}(4261,\cdot)\) \(\chi_{8007}(4732,\cdot)\) \(\chi_{8007}(5545,\cdot)\) \(\chi_{8007}(6958,\cdot)\) \(\chi_{8007}(7015,\cdot)\) \(\chi_{8007}(7486,\cdot)\) \(\chi_{8007}(7558,\cdot)\) \(\chi_{8007}(7900,\cdot)\)

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

\((5339,1414,7855)\) → \((1,e\left(\frac{1}{16}\right),e\left(\frac{7}{12}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 8007 }(292, a) \) \(1\)\(1\)\(e\left(\frac{1}{8}\right)\)\(i\)\(e\left(\frac{43}{48}\right)\)\(e\left(\frac{7}{16}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{1}{48}\right)\)\(e\left(\frac{37}{48}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{9}{16}\right)\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8007 }(292,a) \;\) at \(\;a = \) e.g. 2