Properties

Label 8026.263
Modulus $8026$
Conductor $4013$
Order $59$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8026, base_ring=CyclotomicField(118))
 
M = H._module
 
chi = DirichletCharacter(H, M([20]))
 
pari: [g,chi] = znchar(Mod(263,8026))
 

Basic properties

Modulus: \(8026\)
Conductor: \(4013\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(59\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{4013}(263,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8026.f

\(\chi_{8026}(263,\cdot)\) \(\chi_{8026}(279,\cdot)\) \(\chi_{8026}(301,\cdot)\) \(\chi_{8026}(425,\cdot)\) \(\chi_{8026}(607,\cdot)\) \(\chi_{8026}(655,\cdot)\) \(\chi_{8026}(807,\cdot)\) \(\chi_{8026}(885,\cdot)\) \(\chi_{8026}(957,\cdot)\) \(\chi_{8026}(1143,\cdot)\) \(\chi_{8026}(1527,\cdot)\) \(\chi_{8026}(1577,\cdot)\) \(\chi_{8026}(1803,\cdot)\) \(\chi_{8026}(2127,\cdot)\) \(\chi_{8026}(2145,\cdot)\) \(\chi_{8026}(2163,\cdot)\) \(\chi_{8026}(2247,\cdot)\) \(\chi_{8026}(2315,\cdot)\) \(\chi_{8026}(2885,\cdot)\) \(\chi_{8026}(3027,\cdot)\) \(\chi_{8026}(3565,\cdot)\) \(\chi_{8026}(3647,\cdot)\) \(\chi_{8026}(3719,\cdot)\) \(\chi_{8026}(3805,\cdot)\) \(\chi_{8026}(3899,\cdot)\) \(\chi_{8026}(4053,\cdot)\) \(\chi_{8026}(4067,\cdot)\) \(\chi_{8026}(4189,\cdot)\) \(\chi_{8026}(4215,\cdot)\) \(\chi_{8026}(4311,\cdot)\) ...

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: $\Q(\zeta_{59})$
Fixed field: Number field defined by a degree 59 polynomial

Values on generators

\(4015\) → \(e\left(\frac{10}{59}\right)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(21\)
\( \chi_{ 8026 }(263, a) \) \(1\)\(1\)\(e\left(\frac{45}{59}\right)\)\(e\left(\frac{49}{59}\right)\)\(e\left(\frac{42}{59}\right)\)\(e\left(\frac{31}{59}\right)\)\(e\left(\frac{8}{59}\right)\)\(e\left(\frac{20}{59}\right)\)\(e\left(\frac{35}{59}\right)\)\(e\left(\frac{30}{59}\right)\)\(e\left(\frac{15}{59}\right)\)\(e\left(\frac{28}{59}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8026 }(263,a) \;\) at \(\;a = \) e.g. 2