Properties

Label 4033.1040
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([6,47]))
 
pari: [g,chi] = znchar(Mod(1040,4033))
 

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.io

\(\chi_{4033}(30,\cdot)\) \(\chi_{4033}(67,\cdot)\) \(\chi_{4033}(262,\cdot)\) \(\chi_{4033}(321,\cdot)\) \(\chi_{4033}(696,\cdot)\) \(\chi_{4033}(733,\cdot)\) \(\chi_{4033}(854,\cdot)\) \(\chi_{4033}(987,\cdot)\) \(\chi_{4033}(1040,\cdot)\) \(\chi_{4033}(1175,\cdot)\) \(\chi_{4033}(1188,\cdot)\) \(\chi_{4033}(1261,\cdot)\) \(\chi_{4033}(1468,\cdot)\) \(\chi_{4033}(1730,\cdot)\) \(\chi_{4033}(1797,\cdot)\) \(\chi_{4033}(1801,\cdot)\) \(\chi_{4033}(1952,\cdot)\) \(\chi_{4033}(2167,\cdot)\) \(\chi_{4033}(2361,\cdot)\) \(\chi_{4033}(2409,\cdot)\) \(\chi_{4033}(2435,\cdot)\) \(\chi_{4033}(2445,\cdot)\) \(\chi_{4033}(2557,\cdot)\) \(\chi_{4033}(2685,\cdot)\) \(\chi_{4033}(2852,\cdot)\) \(\chi_{4033}(3062,\cdot)\) \(\chi_{4033}(3092,\cdot)\) \(\chi_{4033}(3284,\cdot)\) \(\chi_{4033}(3444,\cdot)\) \(\chi_{4033}(3540,\cdot)\) ...

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

\((1963,2295)\) → \((e\left(\frac{1}{18}\right),e\left(\frac{47}{108}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4033 }(1040, a) \) \(-1\)\(1\)\(e\left(\frac{31}{36}\right)\)\(e\left(\frac{2}{27}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{19}{54}\right)\)\(e\left(\frac{101}{108}\right)\)\(e\left(\frac{5}{27}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{4}{27}\right)\)\(e\left(\frac{23}{108}\right)\)\(e\left(\frac{85}{108}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4033 }(1040,a) \;\) at \(\;a = \) e.g. 2