Properties

Label 3920.3709
Modulus $3920$
Conductor $3920$
Order $28$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3920, base_ring=CyclotomicField(28))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,21,14,6]))
 
pari: [g,chi] = znchar(Mod(3709,3920))
 

Basic properties

Modulus: \(3920\)
Conductor: \(3920\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(28\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3920.ec

\(\chi_{3920}(69,\cdot)\) \(\chi_{3920}(349,\cdot)\) \(\chi_{3920}(629,\cdot)\) \(\chi_{3920}(909,\cdot)\) \(\chi_{3920}(1189,\cdot)\) \(\chi_{3920}(1749,\cdot)\) \(\chi_{3920}(2029,\cdot)\) \(\chi_{3920}(2309,\cdot)\) \(\chi_{3920}(2589,\cdot)\) \(\chi_{3920}(2869,\cdot)\) \(\chi_{3920}(3149,\cdot)\) \(\chi_{3920}(3709,\cdot)\)

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: \(\Q(\zeta_{28})\)
Fixed field: 28.0.1658791218361497301089292178042702471550650476745587455898419200000000000000.1

Values on generators

\((1471,981,3137,3041)\) → \((1,-i,-1,e\left(\frac{3}{14}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\( \chi_{ 3920 }(3709, a) \) \(-1\)\(1\)\(e\left(\frac{27}{28}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{9}{28}\right)\)\(e\left(\frac{23}{28}\right)\)\(e\left(\frac{6}{7}\right)\)\(-i\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{25}{28}\right)\)\(e\left(\frac{3}{28}\right)\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3920 }(3709,a) \;\) at \(\;a = \) e.g. 2