Properties

Label 8008.1649
Modulus $8008$
Conductor $1001$
Order $12$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8008, base_ring=CyclotomicField(12))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,8,6,7]))
 
pari: [g,chi] = znchar(Mod(1649,8008))
 

Basic properties

Modulus: \(8008\)
Conductor: \(1001\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1001}(648,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8008.kg

\(\chi_{8008}(1649,\cdot)\) \(\chi_{8008}(2529,\cdot)\) \(\chi_{8008}(5345,\cdot)\) \(\chi_{8008}(5609,\cdot)\)

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: \(\Q(\zeta_{12})\)
Fixed field: 12.12.18302790406495149533335357.1

Values on generators

\((6007,4005,3433,4369,4929)\) → \((1,1,e\left(\frac{2}{3}\right),-1,e\left(\frac{7}{12}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(15\)\(17\)\(19\)\(23\)\(25\)\(27\)\(29\)
\( \chi_{ 8008 }(1649, a) \) \(1\)\(1\)\(1\)\(e\left(\frac{7}{12}\right)\)\(1\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{1}{3}\right)\)\(-i\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{6}\right)\)\(1\)\(e\left(\frac{5}{6}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8008 }(1649,a) \;\) at \(\;a = \) e.g. 2