Properties

Label 8021.89
Modulus $8021$
Conductor $8021$
Order $264$
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(264))
 
M = H._module
 
chi = DirichletCharacter(H, M([154,123]))
 
pari: [g,chi] = znchar(Mod(89,8021))
 

Basic properties

Modulus: \(8021\)
Conductor: \(8021\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(264\)
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.dj

\(\chi_{8021}(89,\cdot)\) \(\chi_{8021}(150,\cdot)\) \(\chi_{8021}(228,\cdot)\) \(\chi_{8021}(262,\cdot)\) \(\chi_{8021}(336,\cdot)\) \(\chi_{8021}(513,\cdot)\) \(\chi_{8021}(544,\cdot)\) \(\chi_{8021}(721,\cdot)\) \(\chi_{8021}(899,\cdot)\) \(\chi_{8021}(908,\cdot)\) \(\chi_{8021}(943,\cdot)\) \(\chi_{8021}(1016,\cdot)\) \(\chi_{8021}(1042,\cdot)\) \(\chi_{8021}(1133,\cdot)\) \(\chi_{8021}(1224,\cdot)\) \(\chi_{8021}(1263,\cdot)\) \(\chi_{8021}(1280,\cdot)\) \(\chi_{8021}(1307,\cdot)\) \(\chi_{8021}(1515,\cdot)\) \(\chi_{8021}(1623,\cdot)\) \(\chi_{8021}(1701,\cdot)\) \(\chi_{8021}(1762,\cdot)\) \(\chi_{8021}(1822,\cdot)\) \(\chi_{8021}(1861,\cdot)\) \(\chi_{8021}(1952,\cdot)\) \(\chi_{8021}(2043,\cdot)\) \(\chi_{8021}(2069,\cdot)\) \(\chi_{8021}(2182,\cdot)\) \(\chi_{8021}(2186,\cdot)\) \(\chi_{8021}(2234,\cdot)\) ...

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: $\Q(\zeta_{264})$
Fixed field: Number field defined by a degree 264 polynomial (not computed)

Values on generators

\((6788,2471)\) → \((e\left(\frac{7}{12}\right),e\left(\frac{41}{88}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 8021 }(89, a) \) \(1\)\(1\)\(e\left(\frac{131}{132}\right)\)\(e\left(\frac{211}{264}\right)\)\(e\left(\frac{65}{66}\right)\)\(e\left(\frac{71}{88}\right)\)\(e\left(\frac{19}{24}\right)\)\(e\left(\frac{19}{33}\right)\)\(e\left(\frac{43}{44}\right)\)\(e\left(\frac{79}{132}\right)\)\(e\left(\frac{211}{264}\right)\)\(e\left(\frac{13}{66}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8021 }(89,a) \;\) at \(\;a = \) e.g. 2