Properties

Label 3920.783
Modulus $3920$
Conductor $140$
Order $4$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

Learn more

Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(3920, base_ring=CyclotomicField(4))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([2,0,3,2]))
 
pari: [g,chi] = znchar(Mod(783,3920))
 

Basic properties

Modulus: \(3920\)
Conductor: \(140\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(4\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{140}(83,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3920.v

\(\chi_{3920}(783,\cdot)\) \(\chi_{3920}(1567,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\sqrt{-1}) \)
Fixed field: 4.0.98000.1

Values on generators

\((1471,981,3137,3041)\) → \((-1,1,-i,-1)\)

Values

\(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\(-1\)\(1\)\(i\)\(-1\)\(-1\)\(-i\)\(i\)\(-1\)\(-i\)\(-i\)\(-1\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3920 }(783,a) \;\) at \(\;a = \) e.g. 2