Properties

Label 1309.395
Modulus $1309$
Conductor $1309$
Order $12$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1309, base_ring=CyclotomicField(12))
 
M = H._module
 
chi = DirichletCharacter(H, M([2,6,9]))
 
pari: [g,chi] = znchar(Mod(395,1309))
 

Basic properties

Modulus: \(1309\)
Conductor: \(1309\)
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 1309.bh

\(\chi_{1309}(208,\cdot)\) \(\chi_{1309}(285,\cdot)\) \(\chi_{1309}(395,\cdot)\) \(\chi_{1309}(472,\cdot)\)

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: 12.12.59343998293561502407627433.1

Values on generators

\((1123,596,309)\) → \((e\left(\frac{1}{6}\right),-1,-i)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(12\)\(13\)
\( \chi_{ 1309 }(395, a) \) \(1\)\(1\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{7}{12}\right)\)\(i\)\(1\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{7}{12}\right)\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1309 }(395,a) \;\) at \(\;a = \) e.g. 2