Properties

Label 4033.598
Modulus $4033$
Conductor $4033$
Order $108$
Real no
Primitive yes
Minimal yes
Parity even

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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([81,49]))
 
pari: [g,chi] = znchar(Mod(598,4033))
 

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: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4033.jl

\(\chi_{4033}(6,\cdot)\) \(\chi_{4033}(228,\cdot)\) \(\chi_{4033}(265,\cdot)\) \(\chi_{4033}(290,\cdot)\) \(\chi_{4033}(450,\cdot)\) \(\chi_{4033}(475,\cdot)\) \(\chi_{4033}(487,\cdot)\) \(\chi_{4033}(598,\cdot)\) \(\chi_{4033}(672,\cdot)\) \(\chi_{4033}(968,\cdot)\) \(\chi_{4033}(1079,\cdot)\) \(\chi_{4033}(1338,\cdot)\) \(\chi_{4033}(1474,\cdot)\) \(\chi_{4033}(1486,\cdot)\) \(\chi_{4033}(1585,\cdot)\) \(\chi_{4033}(1659,\cdot)\) \(\chi_{4033}(1918,\cdot)\) \(\chi_{4033}(2004,\cdot)\) \(\chi_{4033}(2029,\cdot)\) \(\chi_{4033}(2115,\cdot)\) \(\chi_{4033}(2374,\cdot)\) \(\chi_{4033}(2448,\cdot)\) \(\chi_{4033}(2547,\cdot)\) \(\chi_{4033}(2559,\cdot)\) \(\chi_{4033}(2695,\cdot)\) \(\chi_{4033}(2954,\cdot)\) \(\chi_{4033}(3065,\cdot)\) \(\chi_{4033}(3361,\cdot)\) \(\chi_{4033}(3435,\cdot)\) \(\chi_{4033}(3546,\cdot)\) ...

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

\((1963,2295)\) → \((-i,e\left(\frac{49}{108}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4033 }(598, a) \) \(1\)\(1\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{5}{54}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{79}{108}\right)\)\(e\left(\frac{19}{27}\right)\)\(e\left(\frac{4}{27}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{5}{27}\right)\)\(e\left(\frac{37}{108}\right)\)\(e\left(\frac{17}{108}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4033 }(598,a) \;\) at \(\;a = \) e.g. 2