Properties

Label 6003.25
Modulus $6003$
Conductor $6003$
Order $231$
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(462))
 
M = H._module
 
chi = DirichletCharacter(H, M([308,42,264]))
 
pari: [g,chi] = znchar(Mod(25,6003))
 

Basic properties

Modulus: \(6003\)
Conductor: \(6003\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(231\)
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.dc

\(\chi_{6003}(16,\cdot)\) \(\chi_{6003}(25,\cdot)\) \(\chi_{6003}(49,\cdot)\) \(\chi_{6003}(52,\cdot)\) \(\chi_{6003}(94,\cdot)\) \(\chi_{6003}(169,\cdot)\) \(\chi_{6003}(223,\cdot)\) \(\chi_{6003}(256,\cdot)\) \(\chi_{6003}(400,\cdot)\) \(\chi_{6003}(430,\cdot)\) \(\chi_{6003}(538,\cdot)\) \(\chi_{6003}(547,\cdot)\) \(\chi_{6003}(616,\cdot)\) \(\chi_{6003}(625,\cdot)\) \(\chi_{6003}(634,\cdot)\) \(\chi_{6003}(745,\cdot)\) \(\chi_{6003}(790,\cdot)\) \(\chi_{6003}(808,\cdot)\) \(\chi_{6003}(832,\cdot)\) \(\chi_{6003}(877,\cdot)\) \(\chi_{6003}(886,\cdot)\) \(\chi_{6003}(922,\cdot)\) \(\chi_{6003}(952,\cdot)\) \(\chi_{6003}(1039,\cdot)\) \(\chi_{6003}(1051,\cdot)\) \(\chi_{6003}(1060,\cdot)\) \(\chi_{6003}(1093,\cdot)\) \(\chi_{6003}(1156,\cdot)\) \(\chi_{6003}(1267,\cdot)\) \(\chi_{6003}(1300,\cdot)\) ...

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: $\Q(\zeta_{231})$
Fixed field: Number field defined by a degree 231 polynomial (not computed)

Values on generators

\((668,3133,4555)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{1}{11}\right),e\left(\frac{4}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 6003 }(25, a) \) \(1\)\(1\)\(e\left(\frac{97}{231}\right)\)\(e\left(\frac{194}{231}\right)\)\(e\left(\frac{230}{231}\right)\)\(e\left(\frac{58}{231}\right)\)\(e\left(\frac{20}{77}\right)\)\(e\left(\frac{32}{77}\right)\)\(e\left(\frac{178}{231}\right)\)\(e\left(\frac{206}{231}\right)\)\(e\left(\frac{155}{231}\right)\)\(e\left(\frac{157}{231}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6003 }(25,a) \;\) at \(\;a = \) e.g. 2