Properties

Label 8281.191
Modulus $8281$
Conductor $637$
Order $21$
Real no
Primitive no
Minimal no
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8281, base_ring=CyclotomicField(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([8,28]))
 
pari: [g,chi] = znchar(Mod(191,8281))
 

Basic properties

Modulus: \(8281\)
Conductor: \(637\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(21\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{637}(191,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8281.bk

\(\chi_{8281}(191,\cdot)\) \(\chi_{8281}(991,\cdot)\) \(\chi_{8281}(1374,\cdot)\) \(\chi_{8281}(2557,\cdot)\) \(\chi_{8281}(3357,\cdot)\) \(\chi_{8281}(3740,\cdot)\) \(\chi_{8281}(4540,\cdot)\) \(\chi_{8281}(4923,\cdot)\) \(\chi_{8281}(5723,\cdot)\) \(\chi_{8281}(6906,\cdot)\) \(\chi_{8281}(7289,\cdot)\) \(\chi_{8281}(8089,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: Number field defined by a degree 21 polynomial

Values on generators

\((1522,3382)\) → \((e\left(\frac{4}{21}\right),e\left(\frac{2}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 8281 }(191, a) \) \(1\)\(1\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{11}{21}\right)\)\(e\left(\frac{10}{21}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{2}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8281 }(191,a) \;\) at \(\;a = \) e.g. 2