Properties

Label 6003.4594
Modulus $6003$
Conductor $6003$
Order $132$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6003, base_ring=CyclotomicField(132))
 
M = H._module
 
chi = DirichletCharacter(H, M([44,42,33]))
 
pari: [g,chi] = znchar(Mod(4594,6003))
 

Basic properties

Modulus: \(6003\)
Conductor: \(6003\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(132\)
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 6003.ct

\(\chi_{6003}(157,\cdot)\) \(\chi_{6003}(481,\cdot)\) \(\chi_{6003}(592,\cdot)\) \(\chi_{6003}(655,\cdot)\) \(\chi_{6003}(916,\cdot)\) \(\chi_{6003}(940,\cdot)\) \(\chi_{6003}(1003,\cdot)\) \(\chi_{6003}(1114,\cdot)\) \(\chi_{6003}(1201,\cdot)\) \(\chi_{6003}(1525,\cdot)\) \(\chi_{6003}(1699,\cdot)\) \(\chi_{6003}(1723,\cdot)\) \(\chi_{6003}(1786,\cdot)\) \(\chi_{6003}(1897,\cdot)\) \(\chi_{6003}(1960,\cdot)\) \(\chi_{6003}(2158,\cdot)\) \(\chi_{6003}(2245,\cdot)\) \(\chi_{6003}(2482,\cdot)\) \(\chi_{6003}(2767,\cdot)\) \(\chi_{6003}(2941,\cdot)\) \(\chi_{6003}(3004,\cdot)\) \(\chi_{6003}(3028,\cdot)\) \(\chi_{6003}(3202,\cdot)\) \(\chi_{6003}(3352,\cdot)\) \(\chi_{6003}(3526,\cdot)\) \(\chi_{6003}(3724,\cdot)\) \(\chi_{6003}(3787,\cdot)\) \(\chi_{6003}(3874,\cdot)\) \(\chi_{6003}(4246,\cdot)\) \(\chi_{6003}(4594,\cdot)\) ...

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: $\Q(\zeta_{132})$
Fixed field: Number field defined by a degree 132 polynomial (not computed)

Values on generators

\((668,3133,4555)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{7}{22}\right),i)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 6003 }(4594, a) \) \(1\)\(1\)\(e\left(\frac{29}{132}\right)\)\(e\left(\frac{29}{66}\right)\)\(e\left(\frac{16}{33}\right)\)\(e\left(\frac{25}{66}\right)\)\(e\left(\frac{29}{44}\right)\)\(e\left(\frac{31}{44}\right)\)\(e\left(\frac{59}{132}\right)\)\(e\left(\frac{41}{66}\right)\)\(e\left(\frac{79}{132}\right)\)\(e\left(\frac{29}{33}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6003 }(4594,a) \;\) at \(\;a = \) e.g. 2