Properties

Label 8027.610
Modulus $8027$
Conductor $8027$
Order $638$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8027, base_ring=CyclotomicField(638))
 
M = H._module
 
chi = DirichletCharacter(H, M([580,363]))
 
pari: [g,chi] = znchar(Mod(610,8027))
 

Basic properties

Modulus: \(8027\)
Conductor: \(8027\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(638\)
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.bm

\(\chi_{8027}(27,\cdot)\) \(\chi_{8027}(36,\cdot)\) \(\chi_{8027}(48,\cdot)\) \(\chi_{8027}(64,\cdot)\) \(\chi_{8027}(75,\cdot)\) \(\chi_{8027}(100,\cdot)\) \(\chi_{8027}(121,\cdot)\) \(\chi_{8027}(223,\cdot)\) \(\chi_{8027}(239,\cdot)\) \(\chi_{8027}(261,\cdot)\) \(\chi_{8027}(282,\cdot)\) \(\chi_{8027}(308,\cdot)\) \(\chi_{8027}(376,\cdot)\) \(\chi_{8027}(386,\cdot)\) \(\chi_{8027}(394,\cdot)\) \(\chi_{8027}(397,\cdot)\) \(\chi_{8027}(409,\cdot)\) \(\chi_{8027}(418,\cdot)\) \(\chi_{8027}(441,\cdot)\) \(\chi_{8027}(449,\cdot)\) \(\chi_{8027}(464,\cdot)\) \(\chi_{8027}(588,\cdot)\) \(\chi_{8027}(610,\cdot)\) \(\chi_{8027}(657,\cdot)\) \(\chi_{8027}(715,\cdot)\) \(\chi_{8027}(725,\cdot)\) \(\chi_{8027}(762,\cdot)\) \(\chi_{8027}(767,\cdot)\) \(\chi_{8027}(784,\cdot)\) \(\chi_{8027}(790,\cdot)\) ...

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 638 polynomial (not computed)

Values on generators

\((350,5935)\) → \((e\left(\frac{10}{11}\right),e\left(\frac{33}{58}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 8027 }(610, a) \) \(1\)\(1\)\(e\left(\frac{247}{638}\right)\)\(e\left(\frac{108}{319}\right)\)\(e\left(\frac{247}{319}\right)\)\(e\left(\frac{180}{319}\right)\)\(e\left(\frac{463}{638}\right)\)\(e\left(\frac{251}{638}\right)\)\(e\left(\frac{103}{638}\right)\)\(e\left(\frac{216}{319}\right)\)\(e\left(\frac{607}{638}\right)\)\(e\left(\frac{83}{638}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8027 }(610,a) \;\) at \(\;a = \) e.g. 2