Properties

Label 8001.139
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([42,63,19]))
 
pari: [g,chi] = znchar(Mod(139,8001))
 

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.na

\(\chi_{8001}(139,\cdot)\) \(\chi_{8001}(601,\cdot)\) \(\chi_{8001}(664,\cdot)\) \(\chi_{8001}(769,\cdot)\) \(\chi_{8001}(853,\cdot)\) \(\chi_{8001}(895,\cdot)\) \(\chi_{8001}(1210,\cdot)\) \(\chi_{8001}(1273,\cdot)\) \(\chi_{8001}(1462,\cdot)\) \(\chi_{8001}(1483,\cdot)\) \(\chi_{8001}(1609,\cdot)\) \(\chi_{8001}(1861,\cdot)\) \(\chi_{8001}(2470,\cdot)\) \(\chi_{8001}(2491,\cdot)\) \(\chi_{8001}(2596,\cdot)\) \(\chi_{8001}(3289,\cdot)\) \(\chi_{8001}(3541,\cdot)\) \(\chi_{8001}(3604,\cdot)\) \(\chi_{8001}(3856,\cdot)\) \(\chi_{8001}(4927,\cdot)\) \(\chi_{8001}(5011,\cdot)\) \(\chi_{8001}(5389,\cdot)\) \(\chi_{8001}(5431,\cdot)\) \(\chi_{8001}(5452,\cdot)\) \(\chi_{8001}(5557,\cdot)\) \(\chi_{8001}(5641,\cdot)\) \(\chi_{8001}(5704,\cdot)\) \(\chi_{8001}(6061,\cdot)\) \(\chi_{8001}(6460,\cdot)\) \(\chi_{8001}(6586,\cdot)\) ...

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

\((3557,1144,7750)\) → \((e\left(\frac{1}{3}\right),-1,e\left(\frac{19}{126}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(19\)
\( \chi_{ 8001 }(139, a) \) \(1\)\(1\)\(e\left(\frac{4}{21}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{37}{63}\right)\)\(e\left(\frac{43}{126}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{29}{126}\right)\)\(e\left(\frac{1}{6}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8001 }(139,a) \;\) at \(\;a = \) e.g. 2