Properties

Label 3920.3383
Modulus $3920$
Conductor $1960$
Order $84$
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(84))
 
M = H._module
 
chi = DirichletCharacter(H, M([42,42,63,52]))
 
pari: [g,chi] = znchar(Mod(3383,3920))
 

Basic properties

Modulus: \(3920\)
Conductor: \(1960\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1960}(443,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3920.fy

\(\chi_{3920}(23,\cdot)\) \(\chi_{3920}(247,\cdot)\) \(\chi_{3920}(487,\cdot)\) \(\chi_{3920}(583,\cdot)\) \(\chi_{3920}(807,\cdot)\) \(\chi_{3920}(823,\cdot)\) \(\chi_{3920}(1143,\cdot)\) \(\chi_{3920}(1367,\cdot)\) \(\chi_{3920}(1383,\cdot)\) \(\chi_{3920}(1607,\cdot)\) \(\chi_{3920}(1703,\cdot)\) \(\chi_{3920}(1927,\cdot)\) \(\chi_{3920}(1943,\cdot)\) \(\chi_{3920}(2167,\cdot)\) \(\chi_{3920}(2263,\cdot)\) \(\chi_{3920}(2487,\cdot)\) \(\chi_{3920}(2503,\cdot)\) \(\chi_{3920}(2727,\cdot)\) \(\chi_{3920}(3047,\cdot)\) \(\chi_{3920}(3063,\cdot)\) \(\chi_{3920}(3287,\cdot)\) \(\chi_{3920}(3383,\cdot)\) \(\chi_{3920}(3623,\cdot)\) \(\chi_{3920}(3847,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((1471,981,3137,3041)\) → \((-1,-1,-i,e\left(\frac{13}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\( \chi_{ 3920 }(3383, a) \) \(1\)\(1\)\(e\left(\frac{73}{84}\right)\)\(e\left(\frac{31}{42}\right)\)\(e\left(\frac{16}{21}\right)\)\(e\left(\frac{5}{28}\right)\)\(e\left(\frac{19}{84}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{23}{84}\right)\)\(e\left(\frac{17}{28}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{5}{6}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3920 }(3383,a) \;\) at \(\;a = \) e.g. 2