Properties

Label 8027.6069
Modulus $8027$
Conductor $8027$
Order $44$
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(44))
 
M = H._module
 
chi = DirichletCharacter(H, M([10,33]))
 
pari: [g,chi] = znchar(Mod(6069,8027))
 

Basic properties

Modulus: \(8027\)
Conductor: \(8027\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(44\)
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.s

\(\chi_{8027}(136,\cdot)\) \(\chi_{8027}(562,\cdot)\) \(\chi_{8027}(911,\cdot)\) \(\chi_{8027}(1183,\cdot)\) \(\chi_{8027}(1532,\cdot)\) \(\chi_{8027}(2307,\cdot)\) \(\chi_{8027}(2656,\cdot)\) \(\chi_{8027}(2928,\cdot)\) \(\chi_{8027}(3005,\cdot)\) \(\chi_{8027}(3277,\cdot)\) \(\chi_{8027}(3354,\cdot)\) \(\chi_{8027}(3626,\cdot)\) \(\chi_{8027}(3975,\cdot)\) \(\chi_{8027}(5448,\cdot)\) \(\chi_{8027}(6069,\cdot)\) \(\chi_{8027}(6146,\cdot)\) \(\chi_{8027}(6767,\cdot)\) \(\chi_{8027}(7193,\cdot)\) \(\chi_{8027}(7542,\cdot)\) \(\chi_{8027}(7814,\cdot)\)

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: \(\Q(\zeta_{44})\)
Fixed field: Number field defined by a degree 44 polynomial

Values on generators

\((350,5935)\) → \((e\left(\frac{5}{22}\right),-i)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 8027 }(6069, a) \) \(1\)\(1\)\(e\left(\frac{9}{44}\right)\)\(e\left(\frac{3}{22}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{15}{44}\right)\)\(e\left(\frac{25}{44}\right)\)\(e\left(\frac{27}{44}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{41}{44}\right)\)\(e\left(\frac{35}{44}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8027 }(6069,a) \;\) at \(\;a = \) e.g. 2