Properties

Label 586971.580
Modulus $586971$
Conductor $586971$
Order $25410$
Real no
Primitive yes
Minimal yes
Parity even

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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([8470,9075,11193]))
 
pari: [g,chi] = znchar(Mod(580,586971))
 

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: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 586971.sg

\(\chi_{586971}(13,\cdot)\) \(\chi_{586971}(139,\cdot)\) \(\chi_{586971}(160,\cdot)\) \(\chi_{586971}(349,\cdot)\) \(\chi_{586971}(580,\cdot)\) \(\chi_{586971}(601,\cdot)\) \(\chi_{586971}(706,\cdot)\) \(\chi_{586971}(853,\cdot)\) \(\chi_{586971}(1042,\cdot)\) \(\chi_{586971}(1084,\cdot)\) \(\chi_{586971}(1168,\cdot)\) \(\chi_{586971}(1294,\cdot)\) \(\chi_{586971}(1399,\cdot)\) \(\chi_{586971}(1525,\cdot)\)

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

\((130439,179686,73207)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{5}{14}\right),e\left(\frac{533}{1210}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(13\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 586971 }(580, a) \) \(1\)\(1\)\(e\left(\frac{1513}{25410}\right)\)\(e\left(\frac{1513}{12705}\right)\)\(e\left(\frac{1907}{25410}\right)\)\(e\left(\frac{1513}{8470}\right)\)\(e\left(\frac{114}{847}\right)\)\(e\left(\frac{6199}{12705}\right)\)\(e\left(\frac{3026}{12705}\right)\)\(e\left(\frac{4097}{4235}\right)\)\(e\left(\frac{92}{605}\right)\)\(e\left(\frac{4933}{25410}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 586971 }(580,a) \;\) at \(\;a = \) e.g. 2