Properties

Label 6003.1393
Modulus $6003$
Conductor $207$
Order $33$
Real no
Primitive no
Minimal yes
Parity even

Related objects

Downloads

Learn more

Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6003, base_ring=CyclotomicField(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([44,42,0]))
 
pari: [g,chi] = znchar(Mod(1393,6003))
 

Basic properties

Modulus: \(6003\)
Conductor: \(207\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(33\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{207}(151,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 6003.bt

\(\chi_{6003}(349,\cdot)\) \(\chi_{6003}(610,\cdot)\) \(\chi_{6003}(1393,\cdot)\) \(\chi_{6003}(1480,\cdot)\) \(\chi_{6003}(1741,\cdot)\) \(\chi_{6003}(1915,\cdot)\) \(\chi_{6003}(2263,\cdot)\) \(\chi_{6003}(2785,\cdot)\) \(\chi_{6003}(3307,\cdot)\) \(\chi_{6003}(3481,\cdot)\) \(\chi_{6003}(3568,\cdot)\) \(\chi_{6003}(3742,\cdot)\) \(\chi_{6003}(4264,\cdot)\) \(\chi_{6003}(4351,\cdot)\) \(\chi_{6003}(4612,\cdot)\) \(\chi_{6003}(4786,\cdot)\) \(\chi_{6003}(5308,\cdot)\) \(\chi_{6003}(5395,\cdot)\) \(\chi_{6003}(5569,\cdot)\) \(\chi_{6003}(5917,\cdot)\)

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: 33.33.70011645999218458416472683122408534303895571350166174758601569.1

Values on generators

\((668,3133,4555)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{7}{11}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 6003 }(1393, a) \) \(1\)\(1\)\(e\left(\frac{31}{33}\right)\)\(e\left(\frac{29}{33}\right)\)\(e\left(\frac{32}{33}\right)\)\(e\left(\frac{25}{33}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{10}{11}\right)\)\(e\left(\frac{13}{33}\right)\)\(e\left(\frac{8}{33}\right)\)\(e\left(\frac{23}{33}\right)\)\(e\left(\frac{25}{33}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 6003 }(1393,a) \;\) at \(\;a = \) e.g. 2