Properties

Label 2320.717
Modulus $2320$
Conductor $2320$
Order $28$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2320, base_ring=CyclotomicField(28))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,21,7,17]))
 
pari: [g,chi] = znchar(Mod(717,2320))
 

Basic properties

Modulus: \(2320\)
Conductor: \(2320\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(28\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 2320.et

\(\chi_{2320}(77,\cdot)\) \(\chi_{2320}(213,\cdot)\) \(\chi_{2320}(453,\cdot)\) \(\chi_{2320}(693,\cdot)\) \(\chi_{2320}(717,\cdot)\) \(\chi_{2320}(797,\cdot)\) \(\chi_{2320}(1013,\cdot)\) \(\chi_{2320}(1597,\cdot)\) \(\chi_{2320}(1813,\cdot)\) \(\chi_{2320}(1893,\cdot)\) \(\chi_{2320}(1917,\cdot)\) \(\chi_{2320}(2157,\cdot)\)

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: Number field defined by a degree 28 polynomial

Values on generators

\((2031,581,1857,321)\) → \((1,-i,i,e\left(\frac{17}{28}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 2320 }(717, a) \) \(1\)\(1\)\(e\left(\frac{1}{28}\right)\)\(e\left(\frac{1}{28}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{13}{14}\right)\)\(1\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{11}{28}\right)\)\(e\left(\frac{3}{28}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2320 }(717,a) \;\) at \(\;a = \) e.g. 2