Properties

Label 3640.499
Modulus $3640$
Conductor $3640$
Order $12$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3640, base_ring=CyclotomicField(12))
 
M = H._module
 
chi = DirichletCharacter(H, M([6,6,6,4,9]))
 
pari: [g,chi] = znchar(Mod(499,3640))
 

Basic properties

Modulus: \(3640\)
Conductor: \(3640\)
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 3640.om

\(\chi_{3640}(499,\cdot)\) \(\chi_{3640}(1019,\cdot)\) \(\chi_{3640}(2179,\cdot)\) \(\chi_{3640}(2699,\cdot)\)

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: Number field defined by a degree 12 polynomial

Values on generators

\((911,1821,1457,521,561)\) → \((-1,-1,-1,e\left(\frac{1}{3}\right),-i)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)\(33\)
\( \chi_{ 3640 }(499, a) \) \(1\)\(1\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{1}{6}\right)\)\(-1\)\(-1\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{5}{12}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3640 }(499,a) \;\) at \(\;a = \) e.g. 2