Properties

Label 8021.43
Modulus $8021$
Conductor $8021$
Order $462$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8021, base_ring=CyclotomicField(462))
 
M = H._module
 
chi = DirichletCharacter(H, M([77,45]))
 
pari: [g,chi] = znchar(Mod(43,8021))
 

Basic properties

Modulus: \(8021\)
Conductor: \(8021\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(462\)
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 8021.dt

\(\chi_{8021}(43,\cdot)\) \(\chi_{8021}(49,\cdot)\) \(\chi_{8021}(121,\cdot)\) \(\chi_{8021}(179,\cdot)\) \(\chi_{8021}(283,\cdot)\) \(\chi_{8021}(309,\cdot)\) \(\chi_{8021}(361,\cdot)\) \(\chi_{8021}(660,\cdot)\) \(\chi_{8021}(738,\cdot)\) \(\chi_{8021}(771,\cdot)\) \(\chi_{8021}(784,\cdot)\) \(\chi_{8021}(836,\cdot)\) \(\chi_{8021}(901,\cdot)\) \(\chi_{8021}(992,\cdot)\) \(\chi_{8021}(1148,\cdot)\) \(\chi_{8021}(1369,\cdot)\) \(\chi_{8021}(1388,\cdot)\) \(\chi_{8021}(1401,\cdot)\) \(\chi_{8021}(1453,\cdot)\) \(\chi_{8021}(1460,\cdot)\) \(\chi_{8021}(1518,\cdot)\) \(\chi_{8021}(1609,\cdot)\) \(\chi_{8021}(1681,\cdot)\) \(\chi_{8021}(1746,\cdot)\) \(\chi_{8021}(1765,\cdot)\) \(\chi_{8021}(1902,\cdot)\) \(\chi_{8021}(1986,\cdot)\) \(\chi_{8021}(2077,\cdot)\) \(\chi_{8021}(2110,\cdot)\) \(\chi_{8021}(2136,\cdot)\) ...

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: $\Q(\zeta_{231})$
Fixed field: Number field defined by a degree 462 polynomial (not computed)

Values on generators

\((6788,2471)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{15}{154}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 8021 }(43, a) \) \(1\)\(1\)\(e\left(\frac{395}{462}\right)\)\(e\left(\frac{353}{462}\right)\)\(e\left(\frac{164}{231}\right)\)\(e\left(\frac{59}{77}\right)\)\(e\left(\frac{13}{21}\right)\)\(e\left(\frac{307}{462}\right)\)\(e\left(\frac{87}{154}\right)\)\(e\left(\frac{122}{231}\right)\)\(e\left(\frac{41}{66}\right)\)\(e\left(\frac{125}{462}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8021 }(43,a) \;\) at \(\;a = \) e.g. 2