Properties

Label 3744.2881
Modulus $3744$
Conductor $13$
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(3744, base_ring=CyclotomicField(4))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([0,0,0,1]))
 
pari: [g,chi] = znchar(Mod(2881,3744))
 

Basic properties

Modulus: \(3744\)
Conductor: \(13\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(4\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{13}(8,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3744.bd

\(\chi_{3744}(577,\cdot)\) \(\chi_{3744}(2881,\cdot)\)

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: 4.0.2197.1

Values on generators

\((703,2341,2081,2017)\) → \((1,1,1,i)\)

Values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 3744 }(2881, a) \) \(-1\)\(1\)\(i\)\(-i\)\(-i\)\(-1\)\(i\)\(-1\)\(-1\)\(1\)\(i\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3744 }(2881,a) \;\) at \(\;a = \) e.g. 2