Properties

Label 10000.23
Modulus $10000$
Conductor $5000$
Order $500$
Real no
Primitive no
Minimal no
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(10000, base_ring=CyclotomicField(500))
 
M = H._module
 
chi = DirichletCharacter(H, M([250,250,431]))
 
pari: [g,chi] = znchar(Mod(23,10000))
 

Basic properties

Modulus: \(10000\)
Conductor: \(5000\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(500\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{5000}(2523,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 10000.cx

\(\chi_{10000}(23,\cdot)\) \(\chi_{10000}(87,\cdot)\) \(\chi_{10000}(103,\cdot)\) \(\chi_{10000}(167,\cdot)\) \(\chi_{10000}(183,\cdot)\) \(\chi_{10000}(247,\cdot)\) \(\chi_{10000}(263,\cdot)\) \(\chi_{10000}(327,\cdot)\) \(\chi_{10000}(423,\cdot)\) \(\chi_{10000}(487,\cdot)\) \(\chi_{10000}(503,\cdot)\) \(\chi_{10000}(567,\cdot)\) \(\chi_{10000}(583,\cdot)\) \(\chi_{10000}(647,\cdot)\) \(\chi_{10000}(663,\cdot)\) \(\chi_{10000}(727,\cdot)\) \(\chi_{10000}(823,\cdot)\) \(\chi_{10000}(887,\cdot)\) \(\chi_{10000}(903,\cdot)\) \(\chi_{10000}(967,\cdot)\) \(\chi_{10000}(983,\cdot)\) \(\chi_{10000}(1047,\cdot)\) \(\chi_{10000}(1063,\cdot)\) \(\chi_{10000}(1127,\cdot)\) \(\chi_{10000}(1223,\cdot)\) \(\chi_{10000}(1287,\cdot)\) \(\chi_{10000}(1303,\cdot)\) \(\chi_{10000}(1367,\cdot)\) \(\chi_{10000}(1383,\cdot)\) \(\chi_{10000}(1447,\cdot)\) ...

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: $\Q(\zeta_{500})$
Fixed field: Number field defined by a degree 500 polynomial (not computed)

Values on generators

\((8751,2501,9377)\) → \((-1,-1,e\left(\frac{431}{500}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 10000 }(23, a) \) \(1\)\(1\)\(e\left(\frac{117}{500}\right)\)\(e\left(\frac{57}{100}\right)\)\(e\left(\frac{117}{250}\right)\)\(e\left(\frac{39}{125}\right)\)\(e\left(\frac{159}{500}\right)\)\(e\left(\frac{63}{500}\right)\)\(e\left(\frac{79}{250}\right)\)\(e\left(\frac{201}{250}\right)\)\(e\left(\frac{11}{500}\right)\)\(e\left(\frac{351}{500}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 10000 }(23,a) \;\) at \(\;a = \) e.g. 2