Properties

Label 3900.743
Modulus $3900$
Conductor $780$
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(3900, base_ring=CyclotomicField(12))
 
M = H._module
 
chi = DirichletCharacter(H, M([6,6,9,1]))
 
pari: [g,chi] = znchar(Mod(743,3900))
 

Basic properties

Modulus: \(3900\)
Conductor: \(780\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(12\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{780}(743,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3900.do

\(\chi_{3900}(743,\cdot)\) \(\chi_{3900}(1307,\cdot)\) \(\chi_{3900}(1943,\cdot)\) \(\chi_{3900}(2507,\cdot)\)

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: 12.12.10451879418023784000000000.1

Values on generators

\((1951,1301,3277,301)\) → \((-1,-1,-i,e\left(\frac{1}{12}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 3900 }(743, a) \) \(1\)\(1\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{1}{3}\right)\)\(i\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{11}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3900 }(743,a) \;\) at \(\;a = \) e.g. 2