Properties

Label 3332.3007
Modulus $3332$
Conductor $476$
Order $24$
Real no
Primitive no
Minimal no
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3332, base_ring=CyclotomicField(24))
 
M = H._module
 
chi = DirichletCharacter(H, M([12,16,9]))
 
pari: [g,chi] = znchar(Mod(3007,3332))
 

Basic properties

Modulus: \(3332\)
Conductor: \(476\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(24\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{476}(151,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3332.bp

\(\chi_{3332}(263,\cdot)\) \(\chi_{3332}(655,\cdot)\) \(\chi_{3332}(1243,\cdot)\) \(\chi_{3332}(1647,\cdot)\) \(\chi_{3332}(2235,\cdot)\) \(\chi_{3332}(2627,\cdot)\) \(\chi_{3332}(3007,\cdot)\) \(\chi_{3332}(3215,\cdot)\)

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: Number field defined by a degree 24 polynomial

Values on generators

\((1667,885,785)\) → \((-1,e\left(\frac{2}{3}\right),e\left(\frac{3}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(19\)\(23\)\(25\)\(27\)
\( \chi_{ 3332 }(3007, a) \) \(-1\)\(1\)\(e\left(\frac{13}{24}\right)\)\(e\left(\frac{5}{24}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{19}{24}\right)\)\(-1\)\(-i\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{11}{24}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{5}{8}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3332 }(3007,a) \;\) at \(\;a = \) e.g. 2