Properties

Label 3920.3317
Modulus $3920$
Conductor $3920$
Order $28$
Real no
Primitive yes
Minimal yes
Parity even

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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([0,7,7,6]))
 
pari: [g,chi] = znchar(Mod(3317,3920))
 

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: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3920.dt

\(\chi_{3920}(13,\cdot)\) \(\chi_{3920}(517,\cdot)\) \(\chi_{3920}(573,\cdot)\) \(\chi_{3920}(1133,\cdot)\) \(\chi_{3920}(1637,\cdot)\) \(\chi_{3920}(1693,\cdot)\) \(\chi_{3920}(2197,\cdot)\) \(\chi_{3920}(2757,\cdot)\) \(\chi_{3920}(2813,\cdot)\) \(\chi_{3920}(3317,\cdot)\) \(\chi_{3920}(3373,\cdot)\) \(\chi_{3920}(3877,\cdot)\)

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: 28.28.129593063934491976647600951409586130589894568495749019992064000000000000000000000.1

Values on generators

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

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\( \chi_{ 3920 }(3317, a) \) \(1\)\(1\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{23}{28}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{17}{28}\right)\)\(-i\)\(e\left(\frac{11}{28}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{3}{28}\right)\)\(-1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3920 }(3317,a) \;\) at \(\;a = \) e.g. 2