Properties

Label 586971.325
Modulus $586971$
Conductor $9317$
Order $3630$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(586971, base_ring=CyclotomicField(3630))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,605,1077]))
 
pari: [g,chi] = znchar(Mod(325,586971))
 

Basic properties

Modulus: \(586971\)
Conductor: \(9317\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(3630\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{9317}(325,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 586971.oz

\(\chi_{586971}(19,\cdot)\) \(\chi_{586971}(325,\cdot)\) \(\chi_{586971}(2224,\cdot)\) \(\chi_{586971}(3412,\cdot)\) \(\chi_{586971}(3988,\cdot)\) \(\chi_{586971}(4429,\cdot)\) \(\chi_{586971}(4870,\cdot)\) \(\chi_{586971}(5617,\cdot)\) \(\chi_{586971}(6058,\cdot)\) \(\chi_{586971}(7075,\cdot)\) \(\chi_{586971}(8263,\cdot)\) \(\chi_{586971}(8839,\cdot)\) \(\chi_{586971}(9280,\cdot)\) \(\chi_{586971}(9721,\cdot)\) \(\chi_{586971}(10027,\cdot)\) \(\chi_{586971}(10468,\cdot)\) \(\chi_{586971}(10909,\cdot)\) \(\chi_{586971}(11926,\cdot)\) \(\chi_{586971}(13114,\cdot)\) \(\chi_{586971}(13690,\cdot)\) \(\chi_{586971}(14131,\cdot)\) \(\chi_{586971}(14572,\cdot)\) \(\chi_{586971}(14878,\cdot)\) \(\chi_{586971}(15319,\cdot)\) \(\chi_{586971}(15760,\cdot)\) \(\chi_{586971}(16777,\cdot)\) \(\chi_{586971}(17965,\cdot)\) \(\chi_{586971}(18541,\cdot)\) \(\chi_{586971}(18982,\cdot)\) \(\chi_{586971}(19423,\cdot)\) ...

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: $\Q(\zeta_{1815})$
Fixed field: Number field defined by a degree 3630 polynomial (not computed)

Values on generators

\((130439,179686,73207)\) → \((1,e\left(\frac{1}{6}\right),e\left(\frac{359}{1210}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(13\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 586971 }(325, a) \) \(1\)\(1\)\(e\left(\frac{2287}{3630}\right)\)\(e\left(\frac{472}{1815}\right)\)\(e\left(\frac{1543}{3630}\right)\)\(e\left(\frac{1077}{1210}\right)\)\(e\left(\frac{20}{363}\right)\)\(e\left(\frac{502}{605}\right)\)\(e\left(\frac{944}{1815}\right)\)\(e\left(\frac{619}{1815}\right)\)\(e\left(\frac{338}{1815}\right)\)\(e\left(\frac{829}{1210}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 586971 }(325,a) \;\) at \(\;a = \) e.g. 2