Properties

Label 8624.1917
Modulus $8624$
Conductor $8624$
Order $140$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8624, base_ring=CyclotomicField(140))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,105,90,112]))
 
pari: [g,chi] = znchar(Mod(1917,8624))
 

Basic properties

Modulus: \(8624\)
Conductor: \(8624\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(140\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 8624.gk

\(\chi_{8624}(69,\cdot)\) \(\chi_{8624}(125,\cdot)\) \(\chi_{8624}(181,\cdot)\) \(\chi_{8624}(405,\cdot)\) \(\chi_{8624}(741,\cdot)\) \(\chi_{8624}(797,\cdot)\) \(\chi_{8624}(1021,\cdot)\) \(\chi_{8624}(1301,\cdot)\) \(\chi_{8624}(1357,\cdot)\) \(\chi_{8624}(1413,\cdot)\) \(\chi_{8624}(1637,\cdot)\) \(\chi_{8624}(1917,\cdot)\) \(\chi_{8624}(1973,\cdot)\) \(\chi_{8624}(2029,\cdot)\) \(\chi_{8624}(2533,\cdot)\) \(\chi_{8624}(2589,\cdot)\) \(\chi_{8624}(2869,\cdot)\) \(\chi_{8624}(3149,\cdot)\) \(\chi_{8624}(3205,\cdot)\) \(\chi_{8624}(3261,\cdot)\) \(\chi_{8624}(3485,\cdot)\) \(\chi_{8624}(3765,\cdot)\) \(\chi_{8624}(3877,\cdot)\) \(\chi_{8624}(4101,\cdot)\) \(\chi_{8624}(4381,\cdot)\) \(\chi_{8624}(4437,\cdot)\) \(\chi_{8624}(4493,\cdot)\) \(\chi_{8624}(4717,\cdot)\) \(\chi_{8624}(5053,\cdot)\) \(\chi_{8624}(5109,\cdot)\) ...

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: $\Q(\zeta_{140})$
Fixed field: Number field defined by a degree 140 polynomial (not computed)

Values on generators

\((5391,6469,7745,3137)\) → \((1,-i,e\left(\frac{9}{14}\right),e\left(\frac{4}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)\(27\)
\( \chi_{ 8624 }(1917, a) \) \(-1\)\(1\)\(e\left(\frac{41}{140}\right)\)\(e\left(\frac{83}{140}\right)\)\(e\left(\frac{41}{70}\right)\)\(e\left(\frac{37}{140}\right)\)\(e\left(\frac{31}{35}\right)\)\(e\left(\frac{19}{70}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{13}{14}\right)\)\(e\left(\frac{13}{70}\right)\)\(e\left(\frac{123}{140}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 8624 }(1917,a) \;\) at \(\;a = \) e.g. 2