Properties

Label 4033.2982
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([93,11]))
 
pari: [g,chi] = znchar(Mod(2982,4033))
 

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.hz

\(\chi_{4033}(56,\cdot)\) \(\chi_{4033}(59,\cdot)\) \(\chi_{4033}(72,\cdot)\) \(\chi_{4033}(91,\cdot)\) \(\chi_{4033}(314,\cdot)\) \(\chi_{4033}(494,\cdot)\) \(\chi_{4033}(498,\cdot)\) \(\chi_{4033}(832,\cdot)\) \(\chi_{4033}(957,\cdot)\) \(\chi_{4033}(967,\cdot)\) \(\chi_{4033}(975,\cdot)\) \(\chi_{4033}(1051,\cdot)\) \(\chi_{4033}(1189,\cdot)\) \(\chi_{4033}(1352,\cdot)\) \(\chi_{4033}(1428,\cdot)\) \(\chi_{4033}(1556,\cdot)\) \(\chi_{4033}(1578,\cdot)\) \(\chi_{4033}(1811,\cdot)\) \(\chi_{4033}(2222,\cdot)\) \(\chi_{4033}(2455,\cdot)\) \(\chi_{4033}(2477,\cdot)\) \(\chi_{4033}(2605,\cdot)\) \(\chi_{4033}(2681,\cdot)\) \(\chi_{4033}(2844,\cdot)\) \(\chi_{4033}(2982,\cdot)\) \(\chi_{4033}(3058,\cdot)\) \(\chi_{4033}(3066,\cdot)\) \(\chi_{4033}(3076,\cdot)\) \(\chi_{4033}(3201,\cdot)\) \(\chi_{4033}(3535,\cdot)\) ...

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

\((1963,2295)\) → \((e\left(\frac{31}{36}\right),e\left(\frac{11}{108}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4033 }(2982, a) \) \(1\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{37}{54}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{59}{108}\right)\)\(e\left(\frac{19}{54}\right)\)\(e\left(\frac{17}{27}\right)\)\(1\)\(e\left(\frac{10}{27}\right)\)\(e\left(\frac{23}{108}\right)\)\(e\left(\frac{31}{108}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4033 }(2982,a) \;\) at \(\;a = \) e.g. 2