Properties

Label 4730.2947
Modulus $4730$
Conductor $2365$
Order $84$
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(84))
 
M = H._module
 
chi = DirichletCharacter(H, M([21,42,32]))
 
pari: [g,chi] = znchar(Mod(2947,4730))
 

Basic properties

Modulus: \(4730\)
Conductor: \(2365\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(84\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2365}(582,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4730.db

\(\chi_{4730}(153,\cdot)\) \(\chi_{4730}(197,\cdot)\) \(\chi_{4730}(483,\cdot)\) \(\chi_{4730}(703,\cdot)\) \(\chi_{4730}(857,\cdot)\) \(\chi_{4730}(1143,\cdot)\) \(\chi_{4730}(1407,\cdot)\) \(\chi_{4730}(1737,\cdot)\) \(\chi_{4730}(1803,\cdot)\) \(\chi_{4730}(2353,\cdot)\) \(\chi_{4730}(2507,\cdot)\) \(\chi_{4730}(2683,\cdot)\) \(\chi_{4730}(2947,\cdot)\) \(\chi_{4730}(3277,\cdot)\) \(\chi_{4730}(3453,\cdot)\) \(\chi_{4730}(3497,\cdot)\) \(\chi_{4730}(3607,\cdot)\) \(\chi_{4730}(3893,\cdot)\) \(\chi_{4730}(3937,\cdot)\) \(\chi_{4730}(4223,\cdot)\) \(\chi_{4730}(4267,\cdot)\) \(\chi_{4730}(4443,\cdot)\) \(\chi_{4730}(4487,\cdot)\) \(\chi_{4730}(4553,\cdot)\)

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: $\Q(\zeta_{84})$
Fixed field: Number field defined by a degree 84 polynomial

Values on generators

\((947,431,1981)\) → \((i,-1,e\left(\frac{8}{21}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 4730 }(2947, a) \) \(1\)\(1\)\(e\left(\frac{11}{84}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{11}{42}\right)\)\(e\left(\frac{37}{84}\right)\)\(e\left(\frac{19}{84}\right)\)\(e\left(\frac{5}{21}\right)\)\(e\left(\frac{3}{14}\right)\)\(e\left(\frac{71}{84}\right)\)\(e\left(\frac{11}{28}\right)\)\(e\left(\frac{13}{21}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4730 }(2947,a) \;\) at \(\;a = \) e.g. 2