Properties

Label 3920.2489
Modulus $3920$
Conductor $1960$
Order $42$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3920, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,21,21,34]))
 
pari: [g,chi] = znchar(Mod(2489,3920))
 

Basic properties

Modulus: \(3920\)
Conductor: \(1960\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1960}(1509,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3920.fa

\(\chi_{3920}(9,\cdot)\) \(\chi_{3920}(249,\cdot)\) \(\chi_{3920}(809,\cdot)\) \(\chi_{3920}(1129,\cdot)\) \(\chi_{3920}(1369,\cdot)\) \(\chi_{3920}(1689,\cdot)\) \(\chi_{3920}(2249,\cdot)\) \(\chi_{3920}(2489,\cdot)\) \(\chi_{3920}(2809,\cdot)\) \(\chi_{3920}(3049,\cdot)\) \(\chi_{3920}(3369,\cdot)\) \(\chi_{3920}(3609,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: Number field defined by a degree 42 polynomial

Values on generators

\((1471,981,3137,3041)\) → \((1,-1,-1,e\left(\frac{17}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\( \chi_{ 3920 }(2489, a) \) \(1\)\(1\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{37}{42}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{11}{42}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{2}{3}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3920 }(2489,a) \;\) at \(\;a = \) e.g. 2