Properties

Label 4730.51
Modulus $4730$
Conductor $473$
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([0,49,65]))
 
pari: [g,chi] = znchar(Mod(51,4730))
 

Basic properties

Modulus: \(4730\)
Conductor: \(473\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(70\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{473}(51,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4730.cr

\(\chi_{4730}(51,\cdot)\) \(\chi_{4730}(151,\cdot)\) \(\chi_{4730}(161,\cdot)\) \(\chi_{4730}(211,\cdot)\) \(\chi_{4730}(371,\cdot)\) \(\chi_{4730}(481,\cdot)\) \(\chi_{4730}(591,\cdot)\) \(\chi_{4730}(1421,\cdot)\) \(\chi_{4730}(1931,\cdot)\) \(\chi_{4730}(2301,\cdot)\) \(\chi_{4730}(2361,\cdot)\) \(\chi_{4730}(2521,\cdot)\) \(\chi_{4730}(2631,\cdot)\) \(\chi_{4730}(2741,\cdot)\) \(\chi_{4730}(2791,\cdot)\) \(\chi_{4730}(3141,\cdot)\) \(\chi_{4730}(3571,\cdot)\) \(\chi_{4730}(4001,\cdot)\) \(\chi_{4730}(4021,\cdot)\) \(\chi_{4730}(4241,\cdot)\) \(\chi_{4730}(4351,\cdot)\) \(\chi_{4730}(4451,\cdot)\) \(\chi_{4730}(4461,\cdot)\) \(\chi_{4730}(4671,\cdot)\)

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

\((947,431,1981)\) → \((1,e\left(\frac{7}{10}\right),e\left(\frac{13}{14}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 4730 }(51, a) \) \(1\)\(1\)\(e\left(\frac{37}{70}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{2}{35}\right)\)\(e\left(\frac{29}{70}\right)\)\(e\left(\frac{41}{70}\right)\)\(e\left(\frac{26}{35}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{41}{70}\right)\)\(e\left(\frac{34}{35}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4730 }(51,a) \;\) at \(\;a = \) e.g. 2