Properties

Label 4730.193
Modulus $4730$
Conductor $2365$
Order $140$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4730, base_ring=CyclotomicField(140))
 
M = H._module
 
chi = DirichletCharacter(H, M([105,126,120]))
 
pari: [g,chi] = znchar(Mod(193,4730))
 

Basic properties

Modulus: \(4730\)
Conductor: \(2365\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(140\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2365}(193,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4730.dg

\(\chi_{4730}(107,\cdot)\) \(\chi_{4730}(127,\cdot)\) \(\chi_{4730}(183,\cdot)\) \(\chi_{4730}(193,\cdot)\) \(\chi_{4730}(293,\cdot)\) \(\chi_{4730}(403,\cdot)\) \(\chi_{4730}(557,\cdot)\) \(\chi_{4730}(563,\cdot)\) \(\chi_{4730}(613,\cdot)\) \(\chi_{4730}(623,\cdot)\) \(\chi_{4730}(723,\cdot)\) \(\chi_{4730}(833,\cdot)\) \(\chi_{4730}(987,\cdot)\) \(\chi_{4730}(1053,\cdot)\) \(\chi_{4730}(1073,\cdot)\) \(\chi_{4730}(1337,\cdot)\) \(\chi_{4730}(1503,\cdot)\) \(\chi_{4730}(1767,\cdot)\) \(\chi_{4730}(1817,\cdot)\) \(\chi_{4730}(1927,\cdot)\) \(\chi_{4730}(1933,\cdot)\) \(\chi_{4730}(2037,\cdot)\) \(\chi_{4730}(2197,\cdot)\) \(\chi_{4730}(2257,\cdot)\) \(\chi_{4730}(2283,\cdot)\) \(\chi_{4730}(2713,\cdot)\) \(\chi_{4730}(2763,\cdot)\) \(\chi_{4730}(2873,\cdot)\) \(\chi_{4730}(2983,\cdot)\) \(\chi_{4730}(3137,\cdot)\) ...

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: $\Q(\zeta_{140})$
Fixed field: Number field defined by a degree 140 polynomial (not computed)

Values on generators

\((947,431,1981)\) → \((-i,e\left(\frac{9}{10}\right),e\left(\frac{6}{7}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 4730 }(193, a) \) \(1\)\(1\)\(e\left(\frac{43}{140}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{43}{70}\right)\)\(e\left(\frac{81}{140}\right)\)\(e\left(\frac{59}{140}\right)\)\(e\left(\frac{17}{35}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{27}{28}\right)\)\(e\left(\frac{129}{140}\right)\)\(e\left(\frac{33}{35}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4730 }(193,a) \;\) at \(\;a = \) e.g. 2