Properties

Label 4730.663
Modulus $4730$
Conductor $2365$
Order $420$
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(420))
 
M = H._module
 
chi = DirichletCharacter(H, M([315,336,290]))
 
pari: [g,chi] = znchar(Mod(663,4730))
 

Basic properties

Modulus: \(4730\)
Conductor: \(2365\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(420\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2365}(663,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4730.do

\(\chi_{4730}(3,\cdot)\) \(\chi_{4730}(147,\cdot)\) \(\chi_{4730}(157,\cdot)\) \(\chi_{4730}(163,\cdot)\) \(\chi_{4730}(313,\cdot)\) \(\chi_{4730}(377,\cdot)\) \(\chi_{4730}(433,\cdot)\) \(\chi_{4730}(493,\cdot)\) \(\chi_{4730}(577,\cdot)\) \(\chi_{4730}(587,\cdot)\) \(\chi_{4730}(663,\cdot)\) \(\chi_{4730}(707,\cdot)\) \(\chi_{4730}(757,\cdot)\) \(\chi_{4730}(807,\cdot)\) \(\chi_{4730}(863,\cdot)\) \(\chi_{4730}(933,\cdot)\) \(\chi_{4730}(1017,\cdot)\) \(\chi_{4730}(1037,\cdot)\) \(\chi_{4730}(1087,\cdot)\) \(\chi_{4730}(1093,\cdot)\) \(\chi_{4730}(1103,\cdot)\) \(\chi_{4730}(1137,\cdot)\) \(\chi_{4730}(1147,\cdot)\) \(\chi_{4730}(1237,\cdot)\) \(\chi_{4730}(1323,\cdot)\) \(\chi_{4730}(1367,\cdot)\) \(\chi_{4730}(1467,\cdot)\) \(\chi_{4730}(1523,\cdot)\) \(\chi_{4730}(1533,\cdot)\)

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: $\Q(\zeta_{420})$
Fixed field: Number field defined by a degree 420 polynomial (not computed)

Values on generators

\((947,431,1981)\) → \((-i,e\left(\frac{4}{5}\right),e\left(\frac{29}{42}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 4730 }(663, a) \) \(1\)\(1\)\(e\left(\frac{143}{420}\right)\)\(e\left(\frac{31}{60}\right)\)\(e\left(\frac{143}{210}\right)\)\(e\left(\frac{61}{420}\right)\)\(e\left(\frac{79}{420}\right)\)\(e\left(\frac{2}{105}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{25}{84}\right)\)\(e\left(\frac{3}{140}\right)\)\(e\left(\frac{43}{105}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4730 }(663,a) \;\) at \(\;a = \) e.g. 2