Properties

Label 8027.629
Modulus $8027$
Conductor $8027$
Order $319$
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(8027, base_ring=CyclotomicField(638))
 
M = H._module
 
chi = DirichletCharacter(H, M([174,66]))
 
pari: [g,chi] = znchar(Mod(629,8027))
 

Basic properties

Modulus: \(8027\)
Conductor: \(8027\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(319\)
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 8027.bi

\(\chi_{8027}(31,\cdot)\) \(\chi_{8027}(41,\cdot)\) \(\chi_{8027}(110,\cdot)\) \(\chi_{8027}(118,\cdot)\) \(\chi_{8027}(210,\cdot)\) \(\chi_{8027}(234,\cdot)\) \(\chi_{8027}(257,\cdot)\) \(\chi_{8027}(269,\cdot)\) \(\chi_{8027}(280,\cdot)\) \(\chi_{8027}(285,\cdot)\) \(\chi_{8027}(289,\cdot)\) \(\chi_{8027}(301,\cdot)\) \(\chi_{8027}(312,\cdot)\) \(\chi_{8027}(380,\cdot)\) \(\chi_{8027}(416,\cdot)\) \(\chi_{8027}(577,\cdot)\) \(\chi_{8027}(583,\cdot)\) \(\chi_{8027}(606,\cdot)\) \(\chi_{8027}(623,\cdot)\) \(\chi_{8027}(629,\cdot)\) \(\chi_{8027}(634,\cdot)\) \(\chi_{8027}(650,\cdot)\) \(\chi_{8027}(653,\cdot)\) \(\chi_{8027}(662,\cdot)\) \(\chi_{8027}(671,\cdot)\) \(\chi_{8027}(729,\cdot)\) \(\chi_{8027}(739,\cdot)\) \(\chi_{8027}(765,\cdot)\) \(\chi_{8027}(786,\cdot)\) \(\chi_{8027}(808,\cdot)\) ...

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: $\Q(\zeta_{319})$
Fixed field: Number field defined by a degree 319 polynomial (not computed)

Values on generators

\((350,5935)\) → \((e\left(\frac{3}{11}\right),e\left(\frac{3}{29}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 8027 }(629, a) \) \(1\)\(1\)\(e\left(\frac{207}{319}\right)\)\(e\left(\frac{17}{319}\right)\)\(e\left(\frac{95}{319}\right)\)\(e\left(\frac{241}{319}\right)\)\(e\left(\frac{224}{319}\right)\)\(e\left(\frac{36}{319}\right)\)\(e\left(\frac{302}{319}\right)\)\(e\left(\frac{34}{319}\right)\)\(e\left(\frac{129}{319}\right)\)\(e\left(\frac{200}{319}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8027 }(629,a) \;\) at \(\;a = \) e.g. 2