Properties

Label 586971.986
Modulus $586971$
Conductor $586971$
Order $25410$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(586971, base_ring=CyclotomicField(25410))
 
M = H._module
 
chi = DirichletCharacter(H, M([21175,16335,10647]))
 
pari: [g,chi] = znchar(Mod(986,586971))
 

Basic properties

Modulus: \(586971\)
Conductor: \(586971\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(25410\)
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 586971.sc

\(\chi_{586971}(41,\cdot)\) \(\chi_{586971}(83,\cdot)\) \(\chi_{586971}(167,\cdot)\) \(\chi_{586971}(272,\cdot)\) \(\chi_{586971}(398,\cdot)\) \(\chi_{586971}(545,\cdot)\) \(\chi_{586971}(776,\cdot)\) \(\chi_{586971}(860,\cdot)\) \(\chi_{586971}(986,\cdot)\) \(\chi_{586971}(1091,\cdot)\) \(\chi_{586971}(1217,\cdot)\) \(\chi_{586971}(1238,\cdot)\) \(\chi_{586971}(1427,\cdot)\)

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: $\Q(\zeta_{12705})$
Fixed field: Number field defined by a degree 25410 polynomial (not computed)

Values on generators

\((130439,179686,73207)\) → \((e\left(\frac{5}{6}\right),e\left(\frac{9}{14}\right),e\left(\frac{507}{1210}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(13\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 586971 }(986, a) \) \(-1\)\(1\)\(e\left(\frac{12281}{12705}\right)\)\(e\left(\frac{11857}{12705}\right)\)\(e\left(\frac{11524}{12705}\right)\)\(e\left(\frac{3811}{4235}\right)\)\(e\left(\frac{740}{847}\right)\)\(e\left(\frac{1396}{12705}\right)\)\(e\left(\frac{11009}{12705}\right)\)\(e\left(\frac{1641}{8470}\right)\)\(e\left(\frac{58}{605}\right)\)\(e\left(\frac{10676}{12705}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 586971 }(986,a) \;\) at \(\;a = \) e.g. 2