Properties

Label 4730.2017
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([35,28,110]))
 
pari: [g,chi] = znchar(Mod(2017,4730))
 

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 \(\chi_{2365}(2017,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4730.df

\(\chi_{4730}(27,\cdot)\) \(\chi_{4730}(113,\cdot)\) \(\chi_{4730}(137,\cdot)\) \(\chi_{4730}(223,\cdot)\) \(\chi_{4730}(247,\cdot)\) \(\chi_{4730}(323,\cdot)\) \(\chi_{4730}(333,\cdot)\) \(\chi_{4730}(383,\cdot)\) \(\chi_{4730}(543,\cdot)\) \(\chi_{4730}(647,\cdot)\) \(\chi_{4730}(653,\cdot)\) \(\chi_{4730}(753,\cdot)\) \(\chi_{4730}(763,\cdot)\) \(\chi_{4730}(973,\cdot)\) \(\chi_{4730}(1083,\cdot)\) \(\chi_{4730}(1193,\cdot)\) \(\chi_{4730}(1527,\cdot)\) \(\chi_{4730}(1587,\cdot)\) \(\chi_{4730}(1593,\cdot)\) \(\chi_{4730}(1747,\cdot)\) \(\chi_{4730}(1857,\cdot)\) \(\chi_{4730}(1967,\cdot)\) \(\chi_{4730}(2017,\cdot)\) \(\chi_{4730}(2447,\cdot)\) \(\chi_{4730}(2473,\cdot)\) \(\chi_{4730}(2533,\cdot)\) \(\chi_{4730}(2693,\cdot)\) \(\chi_{4730}(2797,\cdot)\) \(\chi_{4730}(2803,\cdot)\) \(\chi_{4730}(2913,\cdot)\) ...

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

\((947,431,1981)\) → \((i,e\left(\frac{1}{5}\right),e\left(\frac{11}{14}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 4730 }(2017, a) \) \(1\)\(1\)\(e\left(\frac{19}{140}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{19}{70}\right)\)\(e\left(\frac{13}{140}\right)\)\(e\left(\frac{127}{140}\right)\)\(e\left(\frac{1}{35}\right)\)\(e\left(\frac{2}{7}\right)\)\(e\left(\frac{9}{28}\right)\)\(e\left(\frac{57}{140}\right)\)\(e\left(\frac{4}{35}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4730 }(2017,a) \;\) at \(\;a = \) e.g. 2