Properties

Label 4730.3983
Modulus $4730$
Conductor $215$
Order $28$
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(28))
 
M = H._module
 
chi = DirichletCharacter(H, M([21,0,2]))
 
pari: [g,chi] = znchar(Mod(3983,4730))
 

Basic properties

Modulus: \(4730\)
Conductor: \(215\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(28\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{215}(113,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4730.bu

\(\chi_{4730}(727,\cdot)\) \(\chi_{4730}(1673,\cdot)\) \(\chi_{4730}(1937,\cdot)\) \(\chi_{4730}(2817,\cdot)\) \(\chi_{4730}(2883,\cdot)\) \(\chi_{4730}(3037,\cdot)\) \(\chi_{4730}(3147,\cdot)\) \(\chi_{4730}(3257,\cdot)\) \(\chi_{4730}(3763,\cdot)\) \(\chi_{4730}(3983,\cdot)\) \(\chi_{4730}(4093,\cdot)\) \(\chi_{4730}(4203,\cdot)\)

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: \(\Q(\zeta_{28})\)
Fixed field: 28.28.1407829094312471215113334241722636006626629352569580078125.1

Values on generators

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

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 4730 }(3983, a) \) \(1\)\(1\)\(e\left(\frac{9}{28}\right)\)\(i\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{15}{28}\right)\)\(e\left(\frac{13}{28}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{11}{28}\right)\)\(e\left(\frac{27}{28}\right)\)\(e\left(\frac{3}{7}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4730 }(3983,a) \;\) at \(\;a = \) e.g. 2