Properties

Label 10000.129
Modulus $10000$
Conductor $625$
Order $250$
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(250))
 
M = H._module
 
chi = DirichletCharacter(H, M([0,0,151]))
 
pari: [g,chi] = znchar(Mod(129,10000))
 

Basic properties

Modulus: \(10000\)
Conductor: \(625\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(250\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{625}(129,\cdot)\)
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 10000.cm

\(\chi_{10000}(129,\cdot)\) \(\chi_{10000}(209,\cdot)\) \(\chi_{10000}(289,\cdot)\) \(\chi_{10000}(369,\cdot)\) \(\chi_{10000}(529,\cdot)\) \(\chi_{10000}(609,\cdot)\) \(\chi_{10000}(689,\cdot)\) \(\chi_{10000}(769,\cdot)\) \(\chi_{10000}(929,\cdot)\) \(\chi_{10000}(1009,\cdot)\) \(\chi_{10000}(1089,\cdot)\) \(\chi_{10000}(1169,\cdot)\) \(\chi_{10000}(1329,\cdot)\) \(\chi_{10000}(1409,\cdot)\) \(\chi_{10000}(1489,\cdot)\) \(\chi_{10000}(1569,\cdot)\) \(\chi_{10000}(1729,\cdot)\) \(\chi_{10000}(1809,\cdot)\) \(\chi_{10000}(1889,\cdot)\) \(\chi_{10000}(1969,\cdot)\) \(\chi_{10000}(2129,\cdot)\) \(\chi_{10000}(2209,\cdot)\) \(\chi_{10000}(2289,\cdot)\) \(\chi_{10000}(2369,\cdot)\) \(\chi_{10000}(2529,\cdot)\) \(\chi_{10000}(2609,\cdot)\) \(\chi_{10000}(2689,\cdot)\) \(\chi_{10000}(2769,\cdot)\) \(\chi_{10000}(2929,\cdot)\) \(\chi_{10000}(3009,\cdot)\) ...

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: $\Q(\zeta_{125})$
Fixed field: Number field defined by a degree 250 polynomial (not computed)

Values on generators

\((8751,2501,9377)\) → \((1,1,e\left(\frac{151}{250}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 10000 }(129, a) \) \(1\)\(1\)\(e\left(\frac{157}{250}\right)\)\(e\left(\frac{47}{50}\right)\)\(e\left(\frac{32}{125}\right)\)\(e\left(\frac{63}{125}\right)\)\(e\left(\frac{239}{250}\right)\)\(e\left(\frac{123}{250}\right)\)\(e\left(\frac{59}{125}\right)\)\(e\left(\frac{71}{125}\right)\)\(e\left(\frac{81}{250}\right)\)\(e\left(\frac{221}{250}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 10000 }(129,a) \;\) at \(\;a = \) e.g. 2