Properties

Label 4730.3319
Modulus $4730$
Conductor $2365$
Order $70$
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(70))
 
M = H._module
 
chi = DirichletCharacter(H, M([35,21,65]))
 
pari: [g,chi] = znchar(Mod(3319,4730))
 

Basic properties

Modulus: \(4730\)
Conductor: \(2365\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(70\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{2365}(954,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4730.cx

\(\chi_{4730}(39,\cdot)\) \(\chi_{4730}(409,\cdot)\) \(\chi_{4730}(469,\cdot)\) \(\chi_{4730}(629,\cdot)\) \(\chi_{4730}(739,\cdot)\) \(\chi_{4730}(849,\cdot)\) \(\chi_{4730}(899,\cdot)\) \(\chi_{4730}(1249,\cdot)\) \(\chi_{4730}(1679,\cdot)\) \(\chi_{4730}(2109,\cdot)\) \(\chi_{4730}(2129,\cdot)\) \(\chi_{4730}(2349,\cdot)\) \(\chi_{4730}(2459,\cdot)\) \(\chi_{4730}(2559,\cdot)\) \(\chi_{4730}(2569,\cdot)\) \(\chi_{4730}(2779,\cdot)\) \(\chi_{4730}(2889,\cdot)\) \(\chi_{4730}(2989,\cdot)\) \(\chi_{4730}(2999,\cdot)\) \(\chi_{4730}(3049,\cdot)\) \(\chi_{4730}(3209,\cdot)\) \(\chi_{4730}(3319,\cdot)\) \(\chi_{4730}(3429,\cdot)\) \(\chi_{4730}(4259,\cdot)\)

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: $\Q(\zeta_{35})$
Fixed field: Number field defined by a degree 70 polynomial

Values on generators

\((947,431,1981)\) → \((-1,e\left(\frac{3}{10}\right),e\left(\frac{13}{14}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 4730 }(3319, a) \) \(1\)\(1\)\(e\left(\frac{29}{35}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{23}{35}\right)\)\(e\left(\frac{18}{35}\right)\)\(e\left(\frac{17}{35}\right)\)\(e\left(\frac{19}{35}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{5}{14}\right)\)\(e\left(\frac{17}{35}\right)\)\(e\left(\frac{6}{35}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4730 }(3319,a) \;\) at \(\;a = \) e.g. 2