Properties

Label 1001.142
Modulus $1001$
Conductor $1001$
Order $6$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1001, base_ring=CyclotomicField(6))
 
M = H._module
 
chi = DirichletCharacter(H, M([2,3,3]))
 
pari: [g,chi] = znchar(Mod(142,1001))
 

Basic properties

Modulus: \(1001\)
Conductor: \(1001\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(6\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1001.y

\(\chi_{1001}(142,\cdot)\) \(\chi_{1001}(571,\cdot)\)

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: \(\Q(\sqrt{-3}) \)
Fixed field: 6.0.7021021007.3

Values on generators

\((430,365,925)\) → \((e\left(\frac{1}{3}\right),-1,-1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(12\)\(15\)
\( \chi_{ 1001 }(142, a) \) \(-1\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{6}\right)\)\(1\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{2}{3}\right)\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1001 }(142,a) \;\) at \(\;a = \) e.g. 2