Properties

Label 8400.239
Modulus $8400$
Conductor $300$
Order $10$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8400, base_ring=CyclotomicField(10))
 
M = H._module
 
chi = DirichletCharacter(H, M([5,0,5,3,0]))
 
pari: [g,chi] = znchar(Mod(239,8400))
 

Basic properties

Modulus: \(8400\)
Conductor: \(300\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(10\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{300}(239,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8400.et

\(\chi_{8400}(239,\cdot)\) \(\chi_{8400}(1919,\cdot)\) \(\chi_{8400}(5279,\cdot)\) \(\chi_{8400}(6959,\cdot)\)

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: \(\Q(\zeta_{5})\)
Fixed field: 10.10.189843750000000000.1

Values on generators

\((3151,2101,2801,5377,3601)\) → \((-1,1,-1,e\left(\frac{3}{10}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 8400 }(239, a) \) \(1\)\(1\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{7}{10}\right)\)\(1\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8400 }(239,a) \;\) at \(\;a = \) e.g. 2