Properties

Label 6003.2368
Modulus $6003$
Conductor $667$
Order $28$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6003, base_ring=CyclotomicField(28))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,14,9]))
 
pari: [g,chi] = znchar(Mod(2368,6003))
 

Basic properties

Modulus: \(6003\)
Conductor: \(667\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(28\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{667}(367,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6003.bq

\(\chi_{6003}(298,\cdot)\) \(\chi_{6003}(1540,\cdot)\) \(\chi_{6003}(1954,\cdot)\) \(\chi_{6003}(2161,\cdot)\) \(\chi_{6003}(2368,\cdot)\) \(\chi_{6003}(2782,\cdot)\) \(\chi_{6003}(2989,\cdot)\) \(\chi_{6003}(3403,\cdot)\) \(\chi_{6003}(3610,\cdot)\) \(\chi_{6003}(3817,\cdot)\) \(\chi_{6003}(4231,\cdot)\) \(\chi_{6003}(5473,\cdot)\)

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: \(\Q(\zeta_{28})\)
Fixed field: 28.28.35394489068231220324814698212289719250778220848093751207381.1

Values on generators

\((668,3133,4555)\) → \((1,-1,e\left(\frac{9}{28}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 6003 }(2368, a) \) \(1\)\(1\)\(e\left(\frac{9}{28}\right)\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{27}{28}\right)\)\(e\left(\frac{25}{28}\right)\)\(e\left(\frac{15}{28}\right)\)\(e\left(\frac{11}{14}\right)\)\(e\left(\frac{19}{28}\right)\)\(e\left(\frac{2}{7}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6003 }(2368,a) \;\) at \(\;a = \) e.g. 2