Properties

Label 8001.473
Modulus $8001$
Conductor $8001$
Order $126$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8001, base_ring=CyclotomicField(126))
 
M = H._module
 
chi = DirichletCharacter(H, M([105,84,13]))
 
pari: [g,chi] = znchar(Mod(473,8001))
 

Basic properties

Modulus: \(8001\)
Conductor: \(8001\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(126\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8001.mv

\(\chi_{8001}(473,\cdot)\) \(\chi_{8001}(947,\cdot)\) \(\chi_{8001}(1166,\cdot)\) \(\chi_{8001}(1325,\cdot)\) \(\chi_{8001}(1388,\cdot)\) \(\chi_{8001}(1577,\cdot)\) \(\chi_{8001}(1640,\cdot)\) \(\chi_{8001}(2300,\cdot)\) \(\chi_{8001}(2396,\cdot)\) \(\chi_{8001}(2522,\cdot)\) \(\chi_{8001}(2552,\cdot)\) \(\chi_{8001}(2585,\cdot)\) \(\chi_{8001}(2837,\cdot)\) \(\chi_{8001}(2900,\cdot)\) \(\chi_{8001}(3182,\cdot)\) \(\chi_{8001}(3308,\cdot)\) \(\chi_{8001}(3341,\cdot)\) \(\chi_{8001}(3623,\cdot)\) \(\chi_{8001}(3686,\cdot)\) \(\chi_{8001}(3875,\cdot)\) \(\chi_{8001}(4538,\cdot)\) \(\chi_{8001}(4601,\cdot)\) \(\chi_{8001}(4790,\cdot)\) \(\chi_{8001}(4883,\cdot)\) \(\chi_{8001}(5009,\cdot)\) \(\chi_{8001}(5420,\cdot)\) \(\chi_{8001}(5546,\cdot)\) \(\chi_{8001}(5702,\cdot)\) \(\chi_{8001}(5798,\cdot)\) \(\chi_{8001}(5954,\cdot)\) ...

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: $\Q(\zeta_{63})$
Fixed field: Number field defined by a degree 126 polynomial (not computed)

Values on generators

\((3557,1144,7750)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{2}{3}\right),e\left(\frac{13}{126}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 8001 }(473, a) \) \(1\)\(1\)\(e\left(\frac{25}{42}\right)\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{65}{126}\right)\)\(e\left(\frac{23}{63}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{11}{126}\right)\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8001 }(473,a) \;\) at \(\;a = \) e.g. 2