Properties

Label 8015.59
Modulus $8015$
Conductor $8015$
Order $228$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8015, base_ring=CyclotomicField(228))
 
M = H._module
 
chi = DirichletCharacter(H, M([114,38,65]))
 
pari: [g,chi] = znchar(Mod(59,8015))
 

Basic properties

Modulus: \(8015\)
Conductor: \(8015\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(228\)
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 8015.he

\(\chi_{8015}(59,\cdot)\) \(\chi_{8015}(164,\cdot)\) \(\chi_{8015}(194,\cdot)\) \(\chi_{8015}(264,\cdot)\) \(\chi_{8015}(269,\cdot)\) \(\chi_{8015}(339,\cdot)\) \(\chi_{8015}(614,\cdot)\) \(\chi_{8015}(754,\cdot)\) \(\chi_{8015}(759,\cdot)\) \(\chi_{8015}(789,\cdot)\) \(\chi_{8015}(824,\cdot)\) \(\chi_{8015}(829,\cdot)\) \(\chi_{8015}(1139,\cdot)\) \(\chi_{8015}(1174,\cdot)\) \(\chi_{8015}(1214,\cdot)\) \(\chi_{8015}(1634,\cdot)\) \(\chi_{8015}(1699,\cdot)\) \(\chi_{8015}(1734,\cdot)\) \(\chi_{8015}(1769,\cdot)\) \(\chi_{8015}(1879,\cdot)\) \(\chi_{8015}(1949,\cdot)\) \(\chi_{8015}(1984,\cdot)\) \(\chi_{8015}(2054,\cdot)\) \(\chi_{8015}(2084,\cdot)\) \(\chi_{8015}(2089,\cdot)\) \(\chi_{8015}(2224,\cdot)\) \(\chi_{8015}(2329,\cdot)\) \(\chi_{8015}(2364,\cdot)\) \(\chi_{8015}(2369,\cdot)\) \(\chi_{8015}(2469,\cdot)\) ...

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: $\Q(\zeta_{228})$
Fixed field: Number field defined by a degree 228 polynomial (not computed)

Values on generators

\((3207,4581,4586)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{65}{228}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(11\)\(12\)\(13\)\(16\)
\( \chi_{ 8015 }(59, a) \) \(1\)\(1\)\(e\left(\frac{187}{228}\right)\)\(e\left(\frac{55}{57}\right)\)\(e\left(\frac{73}{114}\right)\)\(e\left(\frac{179}{228}\right)\)\(e\left(\frac{35}{76}\right)\)\(e\left(\frac{53}{57}\right)\)\(e\left(\frac{97}{114}\right)\)\(e\left(\frac{23}{38}\right)\)\(e\left(\frac{27}{76}\right)\)\(e\left(\frac{16}{57}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8015 }(59,a) \;\) at \(\;a = \) e.g. 2