Properties

Label 6003.5509
Modulus $6003$
Conductor $667$
Order $22$
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(22))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,20,11]))
 
pari: [g,chi] = znchar(Mod(5509,6003))
 

Basic properties

Modulus: \(6003\)
Conductor: \(667\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(22\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{667}(173,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6003.bl

\(\chi_{6003}(289,\cdot)\) \(\chi_{6003}(811,\cdot)\) \(\chi_{6003}(2377,\cdot)\) \(\chi_{6003}(2638,\cdot)\) \(\chi_{6003}(3160,\cdot)\) \(\chi_{6003}(3682,\cdot)\) \(\chi_{6003}(4204,\cdot)\) \(\chi_{6003}(4465,\cdot)\) \(\chi_{6003}(5248,\cdot)\) \(\chi_{6003}(5509,\cdot)\)

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

\((668,3133,4555)\) → \((1,e\left(\frac{10}{11}\right),-1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 6003 }(5509, a) \) \(1\)\(1\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{7}{11}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{3}{11}\right)\)\(e\left(\frac{21}{22}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{3}{11}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6003 }(5509,a) \;\) at \(\;a = \) e.g. 2