Properties

Label 4033.942
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([21,65]))
 
pari: [g,chi] = znchar(Mod(942,4033))
 

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.hy

\(\chi_{4033}(24,\cdot)\) \(\chi_{4033}(166,\cdot)\) \(\chi_{4033}(168,\cdot)\) \(\chi_{4033}(466,\cdot)\) \(\chi_{4033}(476,\cdot)\) \(\chi_{4033}(701,\cdot)\) \(\chi_{4033}(942,\cdot)\) \(\chi_{4033}(994,\cdot)\) \(\chi_{4033}(1108,\cdot)\) \(\chi_{4033}(1132,\cdot)\) \(\chi_{4033}(1162,\cdot)\) \(\chi_{4033}(1314,\cdot)\) \(\chi_{4033}(1364,\cdot)\) \(\chi_{4033}(1482,\cdot)\) \(\chi_{4033}(1537,\cdot)\) \(\chi_{4033}(1795,\cdot)\) \(\chi_{4033}(1863,\cdot)\) \(\chi_{4033}(1948,\cdot)\) \(\chi_{4033}(2085,\cdot)\) \(\chi_{4033}(2170,\cdot)\) \(\chi_{4033}(2238,\cdot)\) \(\chi_{4033}(2496,\cdot)\) \(\chi_{4033}(2551,\cdot)\) \(\chi_{4033}(2669,\cdot)\) \(\chi_{4033}(2719,\cdot)\) \(\chi_{4033}(2871,\cdot)\) \(\chi_{4033}(2901,\cdot)\) \(\chi_{4033}(2925,\cdot)\) \(\chi_{4033}(3039,\cdot)\) \(\chi_{4033}(3091,\cdot)\) ...

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

\((1963,2295)\) → \((e\left(\frac{7}{36}\right),e\left(\frac{65}{108}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4033 }(942, a) \) \(1\)\(1\)\(-1\)\(e\left(\frac{19}{54}\right)\)\(1\)\(e\left(\frac{23}{108}\right)\)\(e\left(\frac{23}{27}\right)\)\(e\left(\frac{8}{27}\right)\)\(-1\)\(e\left(\frac{19}{27}\right)\)\(e\left(\frac{77}{108}\right)\)\(e\left(\frac{85}{108}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4033 }(942,a) \;\) at \(\;a = \) e.g. 2