Properties

Label 1037.684
Modulus $1037$
Conductor $1037$
Order $12$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1037, base_ring=CyclotomicField(12))
 
M = H._module
 
chi = DirichletCharacter(H, M([9,8]))
 
pari: [g,chi] = znchar(Mod(684,1037))
 

Basic properties

Modulus: \(1037\)
Conductor: \(1037\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
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 1037.bb

\(\chi_{1037}(13,\cdot)\) \(\chi_{1037}(47,\cdot)\) \(\chi_{1037}(684,\cdot)\) \(\chi_{1037}(718,\cdot)\)

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: 12.12.22734163157293282124804657.1

Values on generators

\((428,307)\) → \((-i,e\left(\frac{2}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1037 }(684, a) \) \(1\)\(1\)\(e\left(\frac{1}{6}\right)\)\(-i\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{11}{12}\right)\)\(-1\)\(-1\)\(e\left(\frac{7}{12}\right)\)\(i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1037 }(684,a) \;\) at \(\;a = \) e.g. 2