Properties

Label 5166.229
Modulus $5166$
Conductor $2583$
Order $120$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5166, base_ring=CyclotomicField(120))
 
M = H._module
 
chi = DirichletCharacter(H, M([40,100,39]))
 
pari: [g,chi] = znchar(Mod(229,5166))
 

Basic properties

Modulus: \(5166\)
Conductor: \(2583\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(120\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2583}(229,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 5166.fq

\(\chi_{5166}(229,\cdot)\) \(\chi_{5166}(481,\cdot)\) \(\chi_{5166}(745,\cdot)\) \(\chi_{5166}(997,\cdot)\) \(\chi_{5166}(1237,\cdot)\) \(\chi_{5166}(1249,\cdot)\) \(\chi_{5166}(1375,\cdot)\) \(\chi_{5166}(1489,\cdot)\) \(\chi_{5166}(1627,\cdot)\) \(\chi_{5166}(1741,\cdot)\) \(\chi_{5166}(1867,\cdot)\) \(\chi_{5166}(1879,\cdot)\) \(\chi_{5166}(2119,\cdot)\) \(\chi_{5166}(2371,\cdot)\) \(\chi_{5166}(2635,\cdot)\) \(\chi_{5166}(2887,\cdot)\) \(\chi_{5166}(3127,\cdot)\) \(\chi_{5166}(3265,\cdot)\) \(\chi_{5166}(3379,\cdot)\) \(\chi_{5166}(3391,\cdot)\) \(\chi_{5166}(3643,\cdot)\) \(\chi_{5166}(3757,\cdot)\) \(\chi_{5166}(3883,\cdot)\) \(\chi_{5166}(4135,\cdot)\) \(\chi_{5166}(4147,\cdot)\) \(\chi_{5166}(4399,\cdot)\) \(\chi_{5166}(4525,\cdot)\) \(\chi_{5166}(4639,\cdot)\) \(\chi_{5166}(4891,\cdot)\) \(\chi_{5166}(4903,\cdot)\) ...

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: $\Q(\zeta_{120})$
Fixed field: Number field defined by a degree 120 polynomial (not computed)

Values on generators

\((2297,2215,3655)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{5}{6}\right),e\left(\frac{13}{40}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(37\)
\( \chi_{ 5166 }(229, a) \) \(1\)\(1\)\(e\left(\frac{59}{60}\right)\)\(e\left(\frac{77}{120}\right)\)\(e\left(\frac{29}{120}\right)\)\(e\left(\frac{67}{120}\right)\)\(e\left(\frac{11}{120}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{73}{120}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{1}{15}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 5166 }(229,a) \;\) at \(\;a = \) e.g. 2