Properties

Label 8033.112
Modulus $8033$
Conductor $8033$
Order $966$
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(966))
 
M = H._module
 
chi = DirichletCharacter(H, M([552,203]))
 
pari: [g,chi] = znchar(Mod(112,8033))
 

Basic properties

Modulus: \(8033\)
Conductor: \(8033\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(966\)
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.ct

\(\chi_{8033}(7,\cdot)\) \(\chi_{8033}(25,\cdot)\) \(\chi_{8033}(36,\cdot)\) \(\chi_{8033}(83,\cdot)\) \(\chi_{8033}(112,\cdot)\) \(\chi_{8033}(123,\cdot)\) \(\chi_{8033}(141,\cdot)\) \(\chi_{8033}(198,\cdot)\) \(\chi_{8033}(210,\cdot)\) \(\chi_{8033}(228,\cdot)\) \(\chi_{8033}(268,\cdot)\) \(\chi_{8033}(284,\cdot)\) \(\chi_{8033}(306,\cdot)\) \(\chi_{8033}(313,\cdot)\) \(\chi_{8033}(339,\cdot)\) \(\chi_{8033}(364,\cdot)\) \(\chi_{8033}(400,\cdot)\) \(\chi_{8033}(487,\cdot)\) \(\chi_{8033}(545,\cdot)\) \(\chi_{8033}(576,\cdot)\) \(\chi_{8033}(616,\cdot)\) \(\chi_{8033}(629,\cdot)\) \(\chi_{8033}(687,\cdot)\) \(\chi_{8033}(741,\cdot)\) \(\chi_{8033}(750,\cdot)\) \(\chi_{8033}(774,\cdot)\) \(\chi_{8033}(803,\cdot)\) \(\chi_{8033}(808,\cdot)\) \(\chi_{8033}(828,\cdot)\) \(\chi_{8033}(865,\cdot)\) ...

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: $\Q(\zeta_{483})$
Fixed field: Number field defined by a degree 966 polynomial (not computed)

Values on generators

\((5541,1944)\) → \((e\left(\frac{4}{7}\right),e\left(\frac{29}{138}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 8033 }(112, a) \) \(1\)\(1\)\(e\left(\frac{149}{322}\right)\)\(e\left(\frac{176}{483}\right)\)\(e\left(\frac{149}{161}\right)\)\(e\left(\frac{755}{966}\right)\)\(e\left(\frac{799}{966}\right)\)\(e\left(\frac{232}{483}\right)\)\(e\left(\frac{125}{322}\right)\)\(e\left(\frac{352}{483}\right)\)\(e\left(\frac{118}{483}\right)\)\(e\left(\frac{731}{966}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8033 }(112,a) \;\) at \(\;a = \) e.g. 2