Properties

Label 3344.263
Modulus $3344$
Conductor $1672$
Order $18$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3344, base_ring=CyclotomicField(18))
 
M = H._module
 
chi = DirichletCharacter(H, M([9,9,9,4]))
 
pari: [g,chi] = znchar(Mod(263,3344))
 

Basic properties

Modulus: \(3344\)
Conductor: \(1672\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(18\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1672}(1099,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3344.cp

\(\chi_{3344}(263,\cdot)\) \(\chi_{3344}(967,\cdot)\) \(\chi_{3344}(1847,\cdot)\) \(\chi_{3344}(2023,\cdot)\) \(\chi_{3344}(2551,\cdot)\) \(\chi_{3344}(3255,\cdot)\)

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: \(\Q(\zeta_{9})\)
Fixed field: Number field defined by a degree 18 polynomial

Values on generators

\((2927,837,2433,705)\) → \((-1,-1,-1,e\left(\frac{2}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(13\)\(15\)\(17\)\(21\)\(23\)\(25\)
\( \chi_{ 3344 }(263, a) \) \(1\)\(1\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{1}{9}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3344 }(263,a) \;\) at \(\;a = \) e.g. 2