Properties

Label 8027.4310
Modulus $8027$
Conductor $8027$
Order $33$
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(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([30,44]))
 
pari: [g,chi] = znchar(Mod(4310,8027))
 

Basic properties

Modulus: \(8027\)
Conductor: \(8027\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(33\)
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.r

\(\chi_{8027}(924,\cdot)\) \(\chi_{8027}(1273,\cdot)\) \(\chi_{8027}(1622,\cdot)\) \(\chi_{8027}(1867,\cdot)\) \(\chi_{8027}(1971,\cdot)\) \(\chi_{8027}(2216,\cdot)\) \(\chi_{8027}(2565,\cdot)\) \(\chi_{8027}(2914,\cdot)\) \(\chi_{8027}(3367,\cdot)\) \(\chi_{8027}(3716,\cdot)\) \(\chi_{8027}(4310,\cdot)\) \(\chi_{8027}(4659,\cdot)\) \(\chi_{8027}(4763,\cdot)\) \(\chi_{8027}(5112,\cdot)\) \(\chi_{8027}(5706,\cdot)\) \(\chi_{8027}(6055,\cdot)\) \(\chi_{8027}(6159,\cdot)\) \(\chi_{8027}(6857,\cdot)\) \(\chi_{8027}(7102,\cdot)\) \(\chi_{8027}(7800,\cdot)\)

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: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 33 polynomial

Values on generators

\((350,5935)\) → \((e\left(\frac{5}{11}\right),e\left(\frac{2}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 8027 }(4310, a) \) \(1\)\(1\)\(e\left(\frac{19}{33}\right)\)\(e\left(\frac{20}{33}\right)\)\(e\left(\frac{5}{33}\right)\)\(e\left(\frac{26}{33}\right)\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{10}{33}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{7}{33}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{1}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8027 }(4310,a) \;\) at \(\;a = \) e.g. 2