Properties

Label 1001.45
Modulus $1001$
Conductor $91$
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(1001, base_ring=CyclotomicField(12))
 
M = H._module
 
chi = DirichletCharacter(H, M([2,0,5]))
 
pari: [g,chi] = znchar(Mod(45,1001))
 

Basic properties

Modulus: \(1001\)
Conductor: \(91\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{91}(45,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1001.ci

\(\chi_{1001}(45,\cdot)\) \(\chi_{1001}(89,\cdot)\) \(\chi_{1001}(782,\cdot)\) \(\chi_{1001}(969,\cdot)\)

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: 12.12.506240953553539690213.2

Values on generators

\((430,365,925)\) → \((e\left(\frac{1}{6}\right),1,e\left(\frac{5}{12}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(12\)\(15\)
\( \chi_{ 1001 }(45, a) \) \(1\)\(1\)\(-i\)\(e\left(\frac{5}{6}\right)\)\(-1\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{7}{12}\right)\)\(i\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{5}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1001 }(45,a) \;\) at \(\;a = \) e.g. 2