Properties

Label 8281.27
Modulus $8281$
Conductor $8281$
Order $182$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8281, base_ring=CyclotomicField(182))
 
M = H._module
 
chi = DirichletCharacter(H, M([13,70]))
 
pari: [g,chi] = znchar(Mod(27,8281))
 

Basic properties

Modulus: \(8281\)
Conductor: \(8281\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(182\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8281.dp

\(\chi_{8281}(27,\cdot)\) \(\chi_{8281}(118,\cdot)\) \(\chi_{8281}(209,\cdot)\) \(\chi_{8281}(300,\cdot)\) \(\chi_{8281}(482,\cdot)\) \(\chi_{8281}(573,\cdot)\) \(\chi_{8281}(664,\cdot)\) \(\chi_{8281}(755,\cdot)\) \(\chi_{8281}(937,\cdot)\) \(\chi_{8281}(1119,\cdot)\) \(\chi_{8281}(1210,\cdot)\) \(\chi_{8281}(1301,\cdot)\) \(\chi_{8281}(1392,\cdot)\) \(\chi_{8281}(1483,\cdot)\) \(\chi_{8281}(1574,\cdot)\) \(\chi_{8281}(1756,\cdot)\) \(\chi_{8281}(1847,\cdot)\) \(\chi_{8281}(1938,\cdot)\) \(\chi_{8281}(2120,\cdot)\) \(\chi_{8281}(2211,\cdot)\) \(\chi_{8281}(2393,\cdot)\) \(\chi_{8281}(2484,\cdot)\) \(\chi_{8281}(2575,\cdot)\) \(\chi_{8281}(2666,\cdot)\) \(\chi_{8281}(2757,\cdot)\) \(\chi_{8281}(2848,\cdot)\) \(\chi_{8281}(3030,\cdot)\) \(\chi_{8281}(3121,\cdot)\) \(\chi_{8281}(3303,\cdot)\) \(\chi_{8281}(3394,\cdot)\) ...

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: $\Q(\zeta_{91})$
Fixed field: Number field defined by a degree 182 polynomial (not computed)

Values on generators

\((1522,3382)\) → \((e\left(\frac{1}{14}\right),e\left(\frac{5}{13}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 8281 }(27, a) \) \(-1\)\(1\)\(e\left(\frac{22}{91}\right)\)\(e\left(\frac{139}{182}\right)\)\(e\left(\frac{44}{91}\right)\)\(e\left(\frac{97}{182}\right)\)\(e\left(\frac{1}{182}\right)\)\(e\left(\frac{66}{91}\right)\)\(e\left(\frac{48}{91}\right)\)\(e\left(\frac{141}{182}\right)\)\(e\left(\frac{43}{91}\right)\)\(e\left(\frac{45}{182}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8281 }(27,a) \;\) at \(\;a = \) e.g. 2