Properties

Label 8021.9
Modulus $8021$
Conductor $8021$
Order $924$
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(924))
 
M = H._module
 
chi = DirichletCharacter(H, M([616,3]))
 
pari: [g,chi] = znchar(Mod(9,8021))
 

Basic properties

Modulus: \(8021\)
Conductor: \(8021\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(924\)
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.dy

\(\chi_{8021}(9,\cdot)\) \(\chi_{8021}(22,\cdot)\) \(\chi_{8021}(74,\cdot)\) \(\chi_{8021}(87,\cdot)\) \(\chi_{8021}(159,\cdot)\) \(\chi_{8021}(230,\cdot)\) \(\chi_{8021}(269,\cdot)\) \(\chi_{8021}(276,\cdot)\) \(\chi_{8021}(289,\cdot)\) \(\chi_{8021}(321,\cdot)\) \(\chi_{8021}(328,\cdot)\) \(\chi_{8021}(341,\cdot)\) \(\chi_{8021}(458,\cdot)\) \(\chi_{8021}(529,\cdot)\) \(\chi_{8021}(705,\cdot)\) \(\chi_{8021}(776,\cdot)\) \(\chi_{8021}(893,\cdot)\) \(\chi_{8021}(906,\cdot)\) \(\chi_{8021}(913,\cdot)\) \(\chi_{8021}(945,\cdot)\) \(\chi_{8021}(958,\cdot)\) \(\chi_{8021}(965,\cdot)\) \(\chi_{8021}(1004,\cdot)\) \(\chi_{8021}(1075,\cdot)\) \(\chi_{8021}(1147,\cdot)\) \(\chi_{8021}(1160,\cdot)\) \(\chi_{8021}(1212,\cdot)\) \(\chi_{8021}(1225,\cdot)\) \(\chi_{8021}(1264,\cdot)\) \(\chi_{8021}(1290,\cdot)\) ...

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: $\Q(\zeta_{924})$
Fixed field: Number field defined by a degree 924 polynomial (not computed)

Values on generators

\((6788,2471)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{1}{308}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 8021 }(9, a) \) \(1\)\(1\)\(e\left(\frac{13}{231}\right)\)\(e\left(\frac{619}{924}\right)\)\(e\left(\frac{26}{231}\right)\)\(e\left(\frac{13}{308}\right)\)\(e\left(\frac{61}{84}\right)\)\(e\left(\frac{367}{462}\right)\)\(e\left(\frac{13}{77}\right)\)\(e\left(\frac{157}{462}\right)\)\(e\left(\frac{13}{132}\right)\)\(e\left(\frac{17}{462}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8021 }(9,a) \;\) at \(\;a = \) e.g. 2