Properties

Label 4033.1243
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([93,89]))
 
pari: [g,chi] = znchar(Mod(1243,4033))
 

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.jx

\(\chi_{4033}(39,\cdot)\) \(\chi_{4033}(200,\cdot)\) \(\chi_{4033}(351,\cdot)\) \(\chi_{4033}(597,\cdot)\) \(\chi_{4033}(607,\cdot)\) \(\chi_{4033}(612,\cdot)\) \(\chi_{4033}(912,\cdot)\) \(\chi_{4033}(1127,\cdot)\) \(\chi_{4033}(1149,\cdot)\) \(\chi_{4033}(1243,\cdot)\) \(\chi_{4033}(1278,\cdot)\) \(\chi_{4033}(1430,\cdot)\) \(\chi_{4033}(1475,\cdot)\) \(\chi_{4033}(1515,\cdot)\) \(\chi_{4033}(1536,\cdot)\) \(\chi_{4033}(1758,\cdot)\) \(\chi_{4033}(1800,\cdot)\) \(\chi_{4033}(1956,\cdot)\) \(\chi_{4033}(2077,\cdot)\) \(\chi_{4033}(2233,\cdot)\) \(\chi_{4033}(2275,\cdot)\) \(\chi_{4033}(2497,\cdot)\) \(\chi_{4033}(2518,\cdot)\) \(\chi_{4033}(2558,\cdot)\) \(\chi_{4033}(2603,\cdot)\) \(\chi_{4033}(2755,\cdot)\) \(\chi_{4033}(2790,\cdot)\) \(\chi_{4033}(2884,\cdot)\) \(\chi_{4033}(2906,\cdot)\) \(\chi_{4033}(3121,\cdot)\) ...

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

\((1963,2295)\) → \((e\left(\frac{31}{36}\right),e\left(\frac{89}{108}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4033 }(1243, a) \) \(1\)\(1\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{13}{54}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{47}{108}\right)\)\(e\left(\frac{2}{27}\right)\)\(e\left(\frac{14}{27}\right)\)\(-1\)\(e\left(\frac{13}{27}\right)\)\(e\left(\frac{29}{108}\right)\)\(e\left(\frac{25}{108}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4033 }(1243,a) \;\) at \(\;a = \) e.g. 2