Properties

Label 3744.3373
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([0,21,16,10]))
 
pari: [g,chi] = znchar(Mod(3373,3744))
 

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.kr

\(\chi_{3744}(301,\cdot)\) \(\chi_{3744}(661,\cdot)\) \(\chi_{3744}(709,\cdot)\) \(\chi_{3744}(1501,\cdot)\) \(\chi_{3744}(2173,\cdot)\) \(\chi_{3744}(2533,\cdot)\) \(\chi_{3744}(2581,\cdot)\) \(\chi_{3744}(3373,\cdot)\)

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: 24.0.58941814124004967164712359225739836985369047739311632559462382829568.1

Values on generators

\((703,2341,2081,2017)\) → \((1,e\left(\frac{7}{8}\right),e\left(\frac{2}{3}\right),e\left(\frac{5}{12}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 3744 }(3373, a) \) \(-1\)\(1\)\(e\left(\frac{23}{24}\right)\)\(1\)\(e\left(\frac{23}{24}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{5}{24}\right)\)\(-i\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{23}{24}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{23}{24}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3744 }(3373,a) \;\) at \(\;a = \) e.g. 2