Properties

Label 4719.230
Modulus $4719$
Conductor $4719$
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(4719, base_ring=CyclotomicField(66))
 
M = H._module
 
chi = DirichletCharacter(H, M([33,21,44]))
 
pari: [g,chi] = znchar(Mod(230,4719))
 

Basic properties

Modulus: \(4719\)
Conductor: \(4719\)
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 4719.cp

\(\chi_{4719}(230,\cdot)\) \(\chi_{4719}(263,\cdot)\) \(\chi_{4719}(659,\cdot)\) \(\chi_{4719}(692,\cdot)\) \(\chi_{4719}(1121,\cdot)\) \(\chi_{4719}(1517,\cdot)\) \(\chi_{4719}(1550,\cdot)\) \(\chi_{4719}(1946,\cdot)\) \(\chi_{4719}(1979,\cdot)\) \(\chi_{4719}(2375,\cdot)\) \(\chi_{4719}(2408,\cdot)\) \(\chi_{4719}(2804,\cdot)\) \(\chi_{4719}(2837,\cdot)\) \(\chi_{4719}(3233,\cdot)\) \(\chi_{4719}(3662,\cdot)\) \(\chi_{4719}(3695,\cdot)\) \(\chi_{4719}(4091,\cdot)\) \(\chi_{4719}(4124,\cdot)\) \(\chi_{4719}(4520,\cdot)\) \(\chi_{4719}(4553,\cdot)\)

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

\((1574,3511,4357)\) → \((-1,e\left(\frac{7}{22}\right),e\left(\frac{2}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 4719 }(230, a) \) \(1\)\(1\)\(e\left(\frac{16}{33}\right)\)\(e\left(\frac{32}{33}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{37}{66}\right)\)\(e\left(\frac{5}{11}\right)\)\(e\left(\frac{35}{66}\right)\)\(e\left(\frac{1}{22}\right)\)\(e\left(\frac{31}{33}\right)\)\(e\left(\frac{14}{33}\right)\)\(e\left(\frac{49}{66}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4719 }(230,a) \;\) at \(\;a = \) e.g. 2