Properties

Label 586971.30973
Modulus $586971$
Conductor $53361$
Order $2310$
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(2310))
 
M = H._module
 
chi = DirichletCharacter(H, M([770,1595,483]))
 
pari: [g,chi] = znchar(Mod(30973,586971))
 

Basic properties

Modulus: \(586971\)
Conductor: \(53361\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(2310\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{53361}(50377,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 586971.on

\(\chi_{586971}(40,\cdot)\) \(\chi_{586971}(481,\cdot)\) \(\chi_{586971}(1564,\cdot)\) \(\chi_{586971}(2635,\cdot)\) \(\chi_{586971}(3379,\cdot)\) \(\chi_{586971}(4450,\cdot)\) \(\chi_{586971}(5848,\cdot)\) \(\chi_{586971}(6289,\cdot)\) \(\chi_{586971}(9187,\cdot)\) \(\chi_{586971}(10258,\cdot)\) \(\chi_{586971}(11002,\cdot)\) \(\chi_{586971}(13912,\cdot)\) \(\chi_{586971}(15286,\cdot)\) \(\chi_{586971}(15727,\cdot)\) \(\chi_{586971}(16810,\cdot)\) \(\chi_{586971}(17881,\cdot)\) \(\chi_{586971}(18625,\cdot)\) \(\chi_{586971}(19696,\cdot)\) \(\chi_{586971}(21094,\cdot)\) \(\chi_{586971}(21535,\cdot)\) \(\chi_{586971}(22909,\cdot)\) \(\chi_{586971}(23350,\cdot)\) \(\chi_{586971}(25504,\cdot)\) \(\chi_{586971}(26248,\cdot)\) \(\chi_{586971}(28717,\cdot)\) \(\chi_{586971}(30532,\cdot)\) \(\chi_{586971}(30973,\cdot)\) \(\chi_{586971}(32056,\cdot)\) \(\chi_{586971}(33127,\cdot)\) \(\chi_{586971}(34942,\cdot)\) ...

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: $\Q(\zeta_{1155})$
Fixed field: Number field defined by a degree 2310 polynomial (not computed)

Values on generators

\((130439,179686,73207)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{29}{42}\right),e\left(\frac{23}{110}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(13\)\(16\)\(17\)\(19\)\(20\)
\( \chi_{ 586971 }(30973, a) \) \(1\)\(1\)\(e\left(\frac{381}{770}\right)\)\(e\left(\frac{381}{385}\right)\)\(e\left(\frac{377}{2310}\right)\)\(e\left(\frac{373}{770}\right)\)\(e\left(\frac{152}{231}\right)\)\(e\left(\frac{659}{1155}\right)\)\(e\left(\frac{377}{385}\right)\)\(e\left(\frac{586}{1155}\right)\)\(e\left(\frac{86}{165}\right)\)\(e\left(\frac{353}{2310}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 586971 }(30973,a) \;\) at \(\;a = \) e.g. 2