Properties

Label 4033.1793
Modulus $4033$
Conductor $4033$
Order $108$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4033, base_ring=CyclotomicField(108))
 
M = H._module
 
chi = DirichletCharacter(H, M([21,80]))
 
pari: [g,chi] = znchar(Mod(1793,4033))
 

Basic properties

Modulus: \(4033\)
Conductor: \(4033\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(108\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4033.js

\(\chi_{4033}(15,\cdot)\) \(\chi_{4033}(187,\cdot)\) \(\chi_{4033}(198,\cdot)\) \(\chi_{4033}(315,\cdot)\) \(\chi_{4033}(348,\cdot)\) \(\chi_{4033}(405,\cdot)\) \(\chi_{4033}(439,\cdot)\) \(\chi_{4033}(461,\cdot)\) \(\chi_{4033}(836,\cdot)\) \(\chi_{4033}(1016,\cdot)\) \(\chi_{4033}(1112,\cdot)\) \(\chi_{4033}(1171,\cdot)\) \(\chi_{4033}(1313,\cdot)\) \(\chi_{4033}(1330,\cdot)\) \(\chi_{4033}(1356,\cdot)\) \(\chi_{4033}(1424,\cdot)\) \(\chi_{4033}(1615,\cdot)\) \(\chi_{4033}(1650,\cdot)\) \(\chi_{4033}(1793,\cdot)\) \(\chi_{4033}(1983,\cdot)\) \(\chi_{4033}(2151,\cdot)\) \(\chi_{4033}(2314,\cdot)\) \(\chi_{4033}(2386,\cdot)\) \(\chi_{4033}(2407,\cdot)\) \(\chi_{4033}(2625,\cdot)\) \(\chi_{4033}(2696,\cdot)\) \(\chi_{4033}(2869,\cdot)\) \(\chi_{4033}(3187,\cdot)\) \(\chi_{4033}(3234,\cdot)\) \(\chi_{4033}(3275,\cdot)\) ...

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: $\Q(\zeta_{108})$
Fixed field: Number field defined by a degree 108 polynomial (not computed)

Values on generators

\((1963,2295)\) → \((e\left(\frac{7}{36}\right),e\left(\frac{20}{27}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4033 }(1793, a) \) \(-1\)\(1\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{31}{54}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{83}{108}\right)\)\(e\left(\frac{107}{108}\right)\)\(e\left(\frac{23}{27}\right)\)\(i\)\(e\left(\frac{4}{27}\right)\)\(e\left(\frac{5}{27}\right)\)\(e\left(\frac{17}{54}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4033 }(1793,a) \;\) at \(\;a = \) e.g. 2