Properties

Label 6069.38
Modulus $6069$
Conductor $357$
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(6069, base_ring=CyclotomicField(12))
 
M = H._module
 
chi = DirichletCharacter(H, M([6,2,9]))
 
pari: [g,chi] = znchar(Mod(38,6069))
 

Basic properties

Modulus: \(6069\)
Conductor: \(357\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{357}(38,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6069.y

\(\chi_{6069}(38,\cdot)\) \(\chi_{6069}(1118,\cdot)\) \(\chi_{6069}(2852,\cdot)\) \(\chi_{6069}(4373,\cdot)\)

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: 12.12.24420144017624194286937.1

Values on generators

\((2024,4336,3760)\) → \((-1,e\left(\frac{1}{6}\right),-i)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(19\)\(20\)
\( \chi_{ 6069 }(38, a) \) \(1\)\(1\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{12}\right)\)\(1\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{5}{12}\right)\)\(-1\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(-i\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6069 }(38,a) \;\) at \(\;a = \) e.g. 2