Properties

Label 1309.375
Modulus $1309$
Conductor $7$
Order $3$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1309, base_ring=CyclotomicField(6))
 
M = H._module
 
chi = DirichletCharacter(H, M([4,0,0]))
 
pari: [g,chi] = znchar(Mod(375,1309))
 

Basic properties

Modulus: \(1309\)
Conductor: \(7\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(3\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{7}(4,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 1309.i

\(\chi_{1309}(375,\cdot)\) \(\chi_{1309}(562,\cdot)\)

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: \(\mathbb{Q}(\zeta_3)\)
Fixed field: \(\Q(\zeta_{7})^+\)

Values on generators

\((1123,596,309)\) → \((e\left(\frac{2}{3}\right),1,1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(12\)\(13\)
\( \chi_{ 1309 }(375, a) \) \(1\)\(1\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(1\)\(1\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 1309 }(375,a) \;\) at \(\;a = \) e.g. 2