Properties

Label 6003.2759
Modulus $6003$
Conductor $6003$
Order $42$
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(42))
 
M = H._module
 
chi = DirichletCharacter(H, M([35,21,3]))
 
pari: [g,chi] = znchar(Mod(2759,6003))
 

Basic properties

Modulus: \(6003\)
Conductor: \(6003\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(42\)
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.ca

\(\chi_{6003}(689,\cdot)\) \(\chi_{6003}(758,\cdot)\) \(\chi_{6003}(1310,\cdot)\) \(\chi_{6003}(1517,\cdot)\) \(\chi_{6003}(1724,\cdot)\) \(\chi_{6003}(2759,\cdot)\) \(\chi_{6003}(3863,\cdot)\) \(\chi_{6003}(4691,\cdot)\) \(\chi_{6003}(5312,\cdot)\) \(\chi_{6003}(5519,\cdot)\) \(\chi_{6003}(5726,\cdot)\) \(\chi_{6003}(5864,\cdot)\)

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: \(\Q(\zeta_{21})\)
Fixed field: Number field defined by a degree 42 polynomial

Values on generators

\((668,3133,4555)\) → \((e\left(\frac{5}{6}\right),-1,e\left(\frac{1}{14}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 6003 }(2759, a) \) \(1\)\(1\)\(e\left(\frac{19}{21}\right)\)\(e\left(\frac{17}{21}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{29}{42}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{1}{7}\right)\)\(e\left(\frac{5}{42}\right)\)\(e\left(\frac{20}{21}\right)\)\(e\left(\frac{25}{42}\right)\)\(e\left(\frac{13}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6003 }(2759,a) \;\) at \(\;a = \) e.g. 2