Properties

Label 4033.3276
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([75,1]))
 
pari: [g,chi] = znchar(Mod(3276,4033))
 

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.ik

\(\chi_{4033}(133,\cdot)\) \(\chi_{4033}(276,\cdot)\) \(\chi_{4033}(309,\cdot)\) \(\chi_{4033}(338,\cdot)\) \(\chi_{4033}(684,\cdot)\) \(\chi_{4033}(716,\cdot)\) \(\chi_{4033}(757,\cdot)\) \(\chi_{4033}(816,\cdot)\) \(\chi_{4033}(944,\cdot)\) \(\chi_{4033}(1031,\cdot)\) \(\chi_{4033}(1134,\cdot)\) \(\chi_{4033}(1186,\cdot)\) \(\chi_{4033}(1350,\cdot)\) \(\chi_{4033}(1704,\cdot)\) \(\chi_{4033}(1754,\cdot)\) \(\chi_{4033}(1796,\cdot)\) \(\chi_{4033}(1892,\cdot)\) \(\chi_{4033}(1976,\cdot)\) \(\chi_{4033}(2057,\cdot)\) \(\chi_{4033}(2141,\cdot)\) \(\chi_{4033}(2237,\cdot)\) \(\chi_{4033}(2279,\cdot)\) \(\chi_{4033}(2329,\cdot)\) \(\chi_{4033}(2683,\cdot)\) \(\chi_{4033}(2847,\cdot)\) \(\chi_{4033}(2899,\cdot)\) \(\chi_{4033}(3002,\cdot)\) \(\chi_{4033}(3089,\cdot)\) \(\chi_{4033}(3217,\cdot)\) \(\chi_{4033}(3276,\cdot)\) ...

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

\((1963,2295)\) → \((e\left(\frac{25}{36}\right),e\left(\frac{1}{108}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4033 }(3276, a) \) \(1\)\(1\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{29}{54}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{73}{108}\right)\)\(e\left(\frac{41}{54}\right)\)\(e\left(\frac{16}{27}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{2}{27}\right)\)\(e\left(\frac{97}{108}\right)\)\(e\left(\frac{65}{108}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4033 }(3276,a) \;\) at \(\;a = \) e.g. 2