Properties

Label 10009.26
Modulus $10009$
Conductor $10009$
Order $139$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(10009, base_ring=CyclotomicField(278))
 
M = H._module
 
chi = DirichletCharacter(H, M([236]))
 
pari: [g,chi] = znchar(Mod(26,10009))
 

Basic properties

Modulus: \(10009\)
Conductor: \(10009\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(139\)
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 10009.m

\(\chi_{10009}(26,\cdot)\) \(\chi_{10009}(51,\cdot)\) \(\chi_{10009}(82,\cdot)\) \(\chi_{10009}(193,\cdot)\) \(\chi_{10009}(195,\cdot)\) \(\chi_{10009}(346,\cdot)\) \(\chi_{10009}(351,\cdot)\) \(\chi_{10009}(385,\cdot)\) \(\chi_{10009}(485,\cdot)\) \(\chi_{10009}(615,\cdot)\) \(\chi_{10009}(676,\cdot)\) \(\chi_{10009}(693,\cdot)\) \(\chi_{10009}(873,\cdot)\) \(\chi_{10009}(1107,\cdot)\) \(\chi_{10009}(1326,\cdot)\) \(\chi_{10009}(1338,\cdot)\) \(\chi_{10009}(1445,\cdot)\) \(\chi_{10009}(1523,\cdot)\) \(\chi_{10009}(1543,\cdot)\) \(\chi_{10009}(1673,\cdot)\) \(\chi_{10009}(1703,\cdot)\) \(\chi_{10009}(1788,\cdot)\) \(\chi_{10009}(2132,\cdot)\) \(\chi_{10009}(2228,\cdot)\) \(\chi_{10009}(2534,\cdot)\) \(\chi_{10009}(2595,\cdot)\) \(\chi_{10009}(2601,\cdot)\) \(\chi_{10009}(2618,\cdot)\) \(\chi_{10009}(2645,\cdot)\) \(\chi_{10009}(2680,\cdot)\) ...

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: $\Q(\zeta_{139})$
Fixed field: Number field defined by a degree 139 polynomial (not computed)

Values on generators

\(11\) → \(e\left(\frac{118}{139}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 10009 }(26, a) \) \(1\)\(1\)\(e\left(\frac{86}{139}\right)\)\(e\left(\frac{90}{139}\right)\)\(e\left(\frac{33}{139}\right)\)\(e\left(\frac{98}{139}\right)\)\(e\left(\frac{37}{139}\right)\)\(e\left(\frac{2}{139}\right)\)\(e\left(\frac{119}{139}\right)\)\(e\left(\frac{41}{139}\right)\)\(e\left(\frac{45}{139}\right)\)\(e\left(\frac{118}{139}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 10009 }(26,a) \;\) at \(\;a = \) e.g. 2