Properties

Label 8033.1831
Modulus $8033$
Conductor $8033$
Order $322$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8033, base_ring=CyclotomicField(322))
 
M = H._module
 
chi = DirichletCharacter(H, M([23,196]))
 
pari: [g,chi] = znchar(Mod(1831,8033))
 

Basic properties

Modulus: \(8033\)
Conductor: \(8033\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(322\)
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 8033.cj

\(\chi_{8033}(236,\cdot)\) \(\chi_{8033}(296,\cdot)\) \(\chi_{8033}(361,\cdot)\) \(\chi_{8033}(441,\cdot)\) \(\chi_{8033}(573,\cdot)\) \(\chi_{8033}(584,\cdot)\) \(\chi_{8033}(709,\cdot)\) \(\chi_{8033}(718,\cdot)\) \(\chi_{8033}(729,\cdot)\) \(\chi_{8033}(767,\cdot)\) \(\chi_{8033}(818,\cdot)\) \(\chi_{8033}(847,\cdot)\) \(\chi_{8033}(850,\cdot)\) \(\chi_{8033}(883,\cdot)\) \(\chi_{8033}(962,\cdot)\) \(\chi_{8033}(995,\cdot)\) \(\chi_{8033}(1049,\cdot)\) \(\chi_{8033}(1095,\cdot)\) \(\chi_{8033}(1124,\cdot)\) \(\chi_{8033}(1135,\cdot)\) \(\chi_{8033}(1309,\cdot)\) \(\chi_{8033}(1311,\cdot)\) \(\chi_{8033}(1372,\cdot)\) \(\chi_{8033}(1401,\cdot)\) \(\chi_{8033}(1454,\cdot)\) \(\chi_{8033}(1542,\cdot)\) \(\chi_{8033}(1588,\cdot)\) \(\chi_{8033}(1658,\cdot)\) \(\chi_{8033}(1746,\cdot)\) \(\chi_{8033}(1831,\cdot)\) ...

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: $\Q(\zeta_{161})$
Fixed field: Number field defined by a degree 322 polynomial (not computed)

Values on generators

\((5541,1944)\) → \((e\left(\frac{1}{14}\right),e\left(\frac{14}{23}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 8033 }(1831, a) \) \(1\)\(1\)\(e\left(\frac{177}{322}\right)\)\(e\left(\frac{255}{322}\right)\)\(e\left(\frac{16}{161}\right)\)\(e\left(\frac{29}{161}\right)\)\(e\left(\frac{55}{161}\right)\)\(e\left(\frac{40}{161}\right)\)\(e\left(\frac{209}{322}\right)\)\(e\left(\frac{94}{161}\right)\)\(e\left(\frac{235}{322}\right)\)\(e\left(\frac{15}{322}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8033 }(1831,a) \;\) at \(\;a = \) e.g. 2