Properties

Label 1120.207
Modulus $1120$
Conductor $280$
Order $12$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1120, base_ring=CyclotomicField(12))
 
M = H._module
 
chi = DirichletCharacter(H, M([6,6,3,8]))
 
pari: [g,chi] = znchar(Mod(207,1120))
 

Basic properties

Modulus: \(1120\)
Conductor: \(280\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{280}(67,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1120.db

\(\chi_{1120}(207,\cdot)\) \(\chi_{1120}(303,\cdot)\) \(\chi_{1120}(527,\cdot)\) \(\chi_{1120}(1103,\cdot)\)

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: 12.12.2951578112000000000.1

Values on generators

\((351,421,897,801)\) → \((-1,-1,i,e\left(\frac{2}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\( \chi_{ 1120 }(207, a) \) \(1\)\(1\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{2}{3}\right)\)\(i\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{7}{12}\right)\)\(i\)\(1\)\(e\left(\frac{1}{6}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1120 }(207,a) \;\) at \(\;a = \) e.g. 2