Properties

Label 8041.1828
Modulus $8041$
Conductor $8041$
Order $280$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8041, base_ring=CyclotomicField(280))
 
M = H._module
 
chi = DirichletCharacter(H, M([28,35,100]))
 
pari: [g,chi] = znchar(Mod(1828,8041))
 

Basic properties

Modulus: \(8041\)
Conductor: \(8041\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(280\)
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 8041.fj

\(\chi_{8041}(2,\cdot)\) \(\chi_{8041}(8,\cdot)\) \(\chi_{8041}(94,\cdot)\) \(\chi_{8041}(151,\cdot)\) \(\chi_{8041}(161,\cdot)\) \(\chi_{8041}(376,\cdot)\) \(\chi_{8041}(457,\cdot)\) \(\chi_{8041}(512,\cdot)\) \(\chi_{8041}(733,\cdot)\) \(\chi_{8041}(739,\cdot)\) \(\chi_{8041}(882,\cdot)\) \(\chi_{8041}(899,\cdot)\) \(\chi_{8041}(954,\cdot)\) \(\chi_{8041}(1097,\cdot)\) \(\chi_{8041}(1107,\cdot)\) \(\chi_{8041}(1249,\cdot)\) \(\chi_{8041}(1317,\cdot)\) \(\chi_{8041}(1403,\cdot)\) \(\chi_{8041}(1470,\cdot)\) \(\chi_{8041}(1613,\cdot)\) \(\chi_{8041}(1623,\cdot)\) \(\chi_{8041}(1630,\cdot)\) \(\chi_{8041}(1685,\cdot)\) \(\chi_{8041}(1828,\cdot)\) \(\chi_{8041}(1845,\cdot)\) \(\chi_{8041}(2048,\cdot)\) \(\chi_{8041}(2195,\cdot)\) \(\chi_{8041}(2263,\cdot)\) \(\chi_{8041}(2361,\cdot)\) \(\chi_{8041}(2416,\cdot)\) ...

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: $\Q(\zeta_{280})$
Fixed field: Number field defined by a degree 280 polynomial (not computed)

Values on generators

\((6580,2366,562)\) → \((e\left(\frac{1}{10}\right),e\left(\frac{1}{8}\right),e\left(\frac{5}{14}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 8041 }(1828, a) \) \(1\)\(1\)\(e\left(\frac{69}{140}\right)\)\(e\left(\frac{79}{280}\right)\)\(e\left(\frac{69}{70}\right)\)\(e\left(\frac{267}{280}\right)\)\(e\left(\frac{31}{40}\right)\)\(e\left(\frac{23}{40}\right)\)\(e\left(\frac{67}{140}\right)\)\(e\left(\frac{79}{140}\right)\)\(e\left(\frac{25}{56}\right)\)\(e\left(\frac{15}{56}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8041 }(1828,a) \;\) at \(\;a = \) e.g. 2