Properties

Label 2001.1172
Modulus $2001$
Conductor $2001$
Order $4$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2001, base_ring=CyclotomicField(4))
 
M = H._module
 
chi = DirichletCharacter(H, M([2,2,1]))
 
pari: [g,chi] = znchar(Mod(1172,2001))
 

Basic properties

Modulus: \(2001\)
Conductor: \(2001\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(4\)
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 2001.i

\(\chi_{2001}(1172,\cdot)\) \(\chi_{2001}(1931,\cdot)\)

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: \(\Q(\sqrt{-1}) \)
Fixed field: 4.0.116116029.2

Values on generators

\((668,1132,553)\) → \((-1,-1,i)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 2001 }(1172, a) \) \(-1\)\(1\)\(-i\)\(-1\)\(-1\)\(-1\)\(i\)\(i\)\(i\)\(-1\)\(i\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2001 }(1172,a) \;\) at \(\;a = \) e.g. 2