Properties

Label 3744.1139
Modulus $3744$
Conductor $3744$
Order $24$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3744, base_ring=CyclotomicField(24))
 
M = H._module
 
chi = DirichletCharacter(H, M([12,21,20,6]))
 
pari: [g,chi] = znchar(Mod(1139,3744))
 

Basic properties

Modulus: \(3744\)
Conductor: \(3744\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(24\)
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 3744.im

\(\chi_{3744}(515,\cdot)\) \(\chi_{3744}(1019,\cdot)\) \(\chi_{3744}(1139,\cdot)\) \(\chi_{3744}(1643,\cdot)\) \(\chi_{3744}(2387,\cdot)\) \(\chi_{3744}(2891,\cdot)\) \(\chi_{3744}(3011,\cdot)\) \(\chi_{3744}(3515,\cdot)\)

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: \(\Q(\zeta_{24})\)
Fixed field: 24.0.167161056827296044968372994547982451448299879797074410465895907328.1

Values on generators

\((703,2341,2081,2017)\) → \((-1,e\left(\frac{7}{8}\right),e\left(\frac{5}{6}\right),i)\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 3744 }(1139, a) \) \(-1\)\(1\)\(e\left(\frac{7}{24}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{11}{24}\right)\)\(-1\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{11}{24}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{5}{8}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3744 }(1139,a) \;\) at \(\;a = \) e.g. 2