Properties

Label 4830.3817
Modulus $4830$
Conductor $805$
Order $12$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4830, base_ring=CyclotomicField(12))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,3,4,6]))
 
pari: [g,chi] = znchar(Mod(3817,4830))
 

Basic properties

Modulus: \(4830\)
Conductor: \(805\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{805}(597,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4830.bt

\(\chi_{4830}(1747,\cdot)\) \(\chi_{4830}(2713,\cdot)\) \(\chi_{4830}(3817,\cdot)\) \(\chi_{4830}(4783,\cdot)\)

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: 12.12.1666791876841970703125.1

Values on generators

\((3221,967,2761,1891)\) → \((1,i,e\left(\frac{1}{3}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(29\)\(31\)\(37\)\(41\)\(43\)\(47\)
\( \chi_{ 4830 }(3817, a) \) \(1\)\(1\)\(e\left(\frac{5}{6}\right)\)\(-i\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{2}{3}\right)\)\(-1\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{5}{12}\right)\)\(1\)\(i\)\(e\left(\frac{11}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4830 }(3817,a) \;\) at \(\;a = \) e.g. 2