Properties

Label 6003.5219
Modulus $6003$
Conductor $2001$
Order $22$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6003, base_ring=CyclotomicField(22))
 
M = H._module
 
chi = DirichletCharacter(H, M([11,13,11]))
 
pari: [g,chi] = znchar(Mod(5219,6003))
 

Basic properties

Modulus: \(6003\)
Conductor: \(2001\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2001}(1217,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6003.bi

\(\chi_{6003}(260,\cdot)\) \(\chi_{6003}(521,\cdot)\) \(\chi_{6003}(2087,\cdot)\) \(\chi_{6003}(2609,\cdot)\) \(\chi_{6003}(3392,\cdot)\) \(\chi_{6003}(3653,\cdot)\) \(\chi_{6003}(4436,\cdot)\) \(\chi_{6003}(4697,\cdot)\) \(\chi_{6003}(5219,\cdot)\) \(\chi_{6003}(5741,\cdot)\)

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: \(\Q(\zeta_{11})\)
Fixed field: Number field defined by a degree 22 polynomial

Values on generators

\((668,3133,4555)\) → \((-1,e\left(\frac{13}{22}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 6003 }(5219, a) \) \(1\)\(1\)\(e\left(\frac{2}{11}\right)\)\(e\left(\frac{4}{11}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{6}{11}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{8}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6003 }(5219,a) \;\) at \(\;a = \) e.g. 2