Properties

Label 2320.2299
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([14,7,14,3]))
 
pari: [g,chi] = znchar(Mod(2299,2320))
 

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.du

\(\chi_{2320}(19,\cdot)\) \(\chi_{2320}(619,\cdot)\) \(\chi_{2320}(659,\cdot)\) \(\chi_{2320}(739,\cdot)\) \(\chi_{2320}(859,\cdot)\) \(\chi_{2320}(1099,\cdot)\) \(\chi_{2320}(1419,\cdot)\) \(\chi_{2320}(1539,\cdot)\) \(\chi_{2320}(1859,\cdot)\) \(\chi_{2320}(2099,\cdot)\) \(\chi_{2320}(2219,\cdot)\) \(\chi_{2320}(2299,\cdot)\)

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

\((2031,581,1857,321)\) → \((-1,i,-1,e\left(\frac{3}{28}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 2320 }(2299, a) \) \(1\)\(1\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{3}{7}\right)\)\(e\left(\frac{5}{28}\right)\)\(-i\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{1}{14}\right)\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{6}{7}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 2320 }(2299,a) \;\) at \(\;a = \) e.g. 2