Properties

Label 3920.1137
Modulus $3920$
Conductor $245$
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([0,0,21,26]))
 
pari: [g,chi] = znchar(Mod(1137,3920))
 

Basic properties

Modulus: \(3920\)
Conductor: \(245\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{245}(157,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 3920.fx

\(\chi_{3920}(17,\cdot)\) \(\chi_{3920}(33,\cdot)\) \(\chi_{3920}(257,\cdot)\) \(\chi_{3920}(353,\cdot)\) \(\chi_{3920}(577,\cdot)\) \(\chi_{3920}(593,\cdot)\) \(\chi_{3920}(817,\cdot)\) \(\chi_{3920}(1137,\cdot)\) \(\chi_{3920}(1153,\cdot)\) \(\chi_{3920}(1377,\cdot)\) \(\chi_{3920}(1473,\cdot)\) \(\chi_{3920}(1713,\cdot)\) \(\chi_{3920}(1937,\cdot)\) \(\chi_{3920}(2033,\cdot)\) \(\chi_{3920}(2257,\cdot)\) \(\chi_{3920}(2497,\cdot)\) \(\chi_{3920}(2593,\cdot)\) \(\chi_{3920}(2817,\cdot)\) \(\chi_{3920}(2833,\cdot)\) \(\chi_{3920}(3153,\cdot)\) \(\chi_{3920}(3377,\cdot)\) \(\chi_{3920}(3393,\cdot)\) \(\chi_{3920}(3617,\cdot)\) \(\chi_{3920}(3713,\cdot)\)

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

\((1471,981,3137,3041)\) → \((1,1,i,e\left(\frac{13}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(19\)\(23\)\(27\)\(29\)\(31\)
\( \chi_{ 3920 }(1137, a) \) \(1\)\(1\)\(e\left(\frac{5}{84}\right)\)\(e\left(\frac{5}{42}\right)\)\(e\left(\frac{8}{21}\right)\)\(e\left(\frac{27}{28}\right)\)\(e\left(\frac{83}{84}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{43}{84}\right)\)\(e\left(\frac{5}{28}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{1}{6}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 3920 }(1137,a) \;\) at \(\;a = \) e.g. 2