Properties

Label 8027.367
Modulus $8027$
Conductor $8027$
Order $348$
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(348))
 
M = H._module
 
chi = DirichletCharacter(H, M([174,53]))
 
pari: [g,chi] = znchar(Mod(367,8027))
 

Basic properties

Modulus: \(8027\)
Conductor: \(8027\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(348\)
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.bk

\(\chi_{8027}(114,\cdot)\) \(\chi_{8027}(137,\cdot)\) \(\chi_{8027}(183,\cdot)\) \(\chi_{8027}(229,\cdot)\) \(\chi_{8027}(252,\cdot)\) \(\chi_{8027}(275,\cdot)\) \(\chi_{8027}(367,\cdot)\) \(\chi_{8027}(505,\cdot)\) \(\chi_{8027}(643,\cdot)\) \(\chi_{8027}(666,\cdot)\) \(\chi_{8027}(827,\cdot)\) \(\chi_{8027}(850,\cdot)\) \(\chi_{8027}(873,\cdot)\) \(\chi_{8027}(919,\cdot)\) \(\chi_{8027}(942,\cdot)\) \(\chi_{8027}(965,\cdot)\) \(\chi_{8027}(988,\cdot)\) \(\chi_{8027}(1034,\cdot)\) \(\chi_{8027}(1080,\cdot)\) \(\chi_{8027}(1264,\cdot)\) \(\chi_{8027}(1333,\cdot)\) \(\chi_{8027}(1356,\cdot)\) \(\chi_{8027}(1586,\cdot)\) \(\chi_{8027}(1632,\cdot)\) \(\chi_{8027}(1655,\cdot)\) \(\chi_{8027}(1701,\cdot)\) \(\chi_{8027}(1747,\cdot)\) \(\chi_{8027}(1816,\cdot)\) \(\chi_{8027}(1862,\cdot)\) \(\chi_{8027}(1885,\cdot)\) ...

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: $\Q(\zeta_{348})$
Fixed field: Number field defined by a degree 348 polynomial (not computed)

Values on generators

\((350,5935)\) → \((-1,e\left(\frac{53}{348}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 8027 }(367, a) \) \(1\)\(1\)\(e\left(\frac{53}{348}\right)\)\(e\left(\frac{167}{174}\right)\)\(e\left(\frac{53}{174}\right)\)\(e\left(\frac{28}{87}\right)\)\(e\left(\frac{13}{116}\right)\)\(e\left(\frac{245}{348}\right)\)\(e\left(\frac{53}{116}\right)\)\(e\left(\frac{80}{87}\right)\)\(e\left(\frac{55}{116}\right)\)\(e\left(\frac{81}{116}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8027 }(367,a) \;\) at \(\;a = \) e.g. 2