Properties

Label 6003.2290
Modulus $6003$
Conductor $6003$
Order $66$
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(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([22,42,33]))
 
pari: [g,chi] = znchar(Mod(2290,6003))
 

Basic properties

Modulus: \(6003\)
Conductor: \(6003\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
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.ci

\(\chi_{6003}(202,\cdot)\) \(\chi_{6003}(376,\cdot)\) \(\chi_{6003}(463,\cdot)\) \(\chi_{6003}(637,\cdot)\) \(\chi_{6003}(1159,\cdot)\) \(\chi_{6003}(1246,\cdot)\) \(\chi_{6003}(1507,\cdot)\) \(\chi_{6003}(1681,\cdot)\) \(\chi_{6003}(2203,\cdot)\) \(\chi_{6003}(2290,\cdot)\) \(\chi_{6003}(2464,\cdot)\) \(\chi_{6003}(2812,\cdot)\) \(\chi_{6003}(3247,\cdot)\) \(\chi_{6003}(3508,\cdot)\) \(\chi_{6003}(4291,\cdot)\) \(\chi_{6003}(4378,\cdot)\) \(\chi_{6003}(4639,\cdot)\) \(\chi_{6003}(4813,\cdot)\) \(\chi_{6003}(5161,\cdot)\) \(\chi_{6003}(5683,\cdot)\)

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: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 66 polynomial

Values on generators

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

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 6003 }(2290, a) \) \(1\)\(1\)\(e\left(\frac{7}{66}\right)\)\(e\left(\frac{7}{33}\right)\)\(e\left(\frac{10}{33}\right)\)\(e\left(\frac{14}{33}\right)\)\(e\left(\frac{7}{22}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{37}{66}\right)\)\(e\left(\frac{19}{33}\right)\)\(e\left(\frac{35}{66}\right)\)\(e\left(\frac{14}{33}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6003 }(2290,a) \;\) at \(\;a = \) e.g. 2