Properties

Label 6003.4757
Modulus $6003$
Conductor $207$
Order $66$
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(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([55,45,0]))
 
pari: [g,chi] = znchar(Mod(4757,6003))
 

Basic properties

Modulus: \(6003\)
Conductor: \(207\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(66\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{207}(203,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6003.ch

\(\chi_{6003}(320,\cdot)\) \(\chi_{6003}(842,\cdot)\) \(\chi_{6003}(1190,\cdot)\) \(\chi_{6003}(1364,\cdot)\) \(\chi_{6003}(1625,\cdot)\) \(\chi_{6003}(1712,\cdot)\) \(\chi_{6003}(2495,\cdot)\) \(\chi_{6003}(2756,\cdot)\) \(\chi_{6003}(3191,\cdot)\) \(\chi_{6003}(3539,\cdot)\) \(\chi_{6003}(3713,\cdot)\) \(\chi_{6003}(3800,\cdot)\) \(\chi_{6003}(4322,\cdot)\) \(\chi_{6003}(4496,\cdot)\) \(\chi_{6003}(4757,\cdot)\) \(\chi_{6003}(4844,\cdot)\) \(\chi_{6003}(5366,\cdot)\) \(\chi_{6003}(5540,\cdot)\) \(\chi_{6003}(5627,\cdot)\) \(\chi_{6003}(5801,\cdot)\)

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

\((668,3133,4555)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{15}{22}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 6003 }(4757, a) \) \(1\)\(1\)\(e\left(\frac{13}{66}\right)\)\(e\left(\frac{13}{33}\right)\)\(e\left(\frac{28}{33}\right)\)\(e\left(\frac{19}{66}\right)\)\(e\left(\frac{13}{22}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{32}{33}\right)\)\(e\left(\frac{7}{33}\right)\)\(e\left(\frac{16}{33}\right)\)\(e\left(\frac{26}{33}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6003 }(4757,a) \;\) at \(\;a = \) e.g. 2