Properties

Label 4851.8
Modulus $4851$
Conductor $1617$
Order $70$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4851, base_ring=CyclotomicField(70))
 
M = H._module
 
chi = DirichletCharacter(H, M([35,60,21]))
 
pari: [g,chi] = znchar(Mod(8,4851))
 

Basic properties

Modulus: \(4851\)
Conductor: \(1617\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(70\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{1617}(8,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 4851.ew

\(\chi_{4851}(8,\cdot)\) \(\chi_{4851}(134,\cdot)\) \(\chi_{4851}(260,\cdot)\) \(\chi_{4851}(512,\cdot)\) \(\chi_{4851}(701,\cdot)\) \(\chi_{4851}(827,\cdot)\) \(\chi_{4851}(953,\cdot)\) \(\chi_{4851}(1205,\cdot)\) \(\chi_{4851}(1394,\cdot)\) \(\chi_{4851}(1646,\cdot)\) \(\chi_{4851}(1898,\cdot)\) \(\chi_{4851}(2087,\cdot)\) \(\chi_{4851}(2213,\cdot)\) \(\chi_{4851}(2339,\cdot)\) \(\chi_{4851}(2591,\cdot)\) \(\chi_{4851}(2780,\cdot)\) \(\chi_{4851}(2906,\cdot)\) \(\chi_{4851}(3032,\cdot)\) \(\chi_{4851}(3473,\cdot)\) \(\chi_{4851}(3599,\cdot)\) \(\chi_{4851}(3977,\cdot)\) \(\chi_{4851}(4292,\cdot)\) \(\chi_{4851}(4418,\cdot)\) \(\chi_{4851}(4670,\cdot)\)

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

\((4313,199,442)\) → \((-1,e\left(\frac{6}{7}\right),e\left(\frac{3}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(13\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 4851 }(8, a) \) \(1\)\(1\)\(e\left(\frac{3}{35}\right)\)\(e\left(\frac{6}{35}\right)\)\(e\left(\frac{39}{70}\right)\)\(e\left(\frac{9}{35}\right)\)\(e\left(\frac{9}{14}\right)\)\(e\left(\frac{41}{70}\right)\)\(e\left(\frac{12}{35}\right)\)\(e\left(\frac{22}{35}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{51}{70}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 4851 }(8,a) \;\) at \(\;a = \) e.g. 2