Properties

Label 4176.19
Modulus $4176$
Conductor $464$
Order $28$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4176, base_ring=CyclotomicField(28))
 
M = H._module
 
chi = DirichletCharacter(H, M([14,21,0,9]))
 
pari: [g,chi] = znchar(Mod(19,4176))
 

Basic properties

Modulus: \(4176\)
Conductor: \(464\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(28\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{464}(19,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4176.dy

\(\chi_{4176}(19,\cdot)\) \(\chi_{4176}(739,\cdot)\) \(\chi_{4176}(955,\cdot)\) \(\chi_{4176}(1099,\cdot)\) \(\chi_{4176}(1171,\cdot)\) \(\chi_{4176}(2251,\cdot)\) \(\chi_{4176}(2323,\cdot)\) \(\chi_{4176}(2467,\cdot)\) \(\chi_{4176}(2683,\cdot)\) \(\chi_{4176}(3403,\cdot)\) \(\chi_{4176}(3691,\cdot)\) \(\chi_{4176}(3907,\cdot)\)

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: \(\Q(\zeta_{28})\)
Fixed field: 28.28.461376647922644386311538110285539877693458386764746599325237248.1

Values on generators

\((1567,1045,929,4033)\) → \((-1,-i,1,e\left(\frac{9}{28}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(31\)\(35\)
\( \chi_{ 4176 }(19, a) \) \(1\)\(1\)\(e\left(\frac{23}{28}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{1}{28}\right)\)\(-i\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{23}{28}\right)\)\(e\left(\frac{19}{28}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4176 }(19,a) \;\) at \(\;a = \) e.g. 2