Properties

Label 625.4
Modulus $625$
Conductor $625$
Order $250$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: PariGP / SageMath
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(625, base_ring=CyclotomicField(250))
 
M = H._module
 
chi = DirichletCharacter(H, M([1]))
 
pari: [g,chi] = znchar(Mod(4,625))
 

Basic properties

Modulus: \(625\)
Conductor: \(625\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(250\)
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 625.k

\(\chi_{625}(4,\cdot)\) \(\chi_{625}(9,\cdot)\) \(\chi_{625}(14,\cdot)\) \(\chi_{625}(19,\cdot)\) \(\chi_{625}(29,\cdot)\) \(\chi_{625}(34,\cdot)\) \(\chi_{625}(39,\cdot)\) \(\chi_{625}(44,\cdot)\) \(\chi_{625}(54,\cdot)\) \(\chi_{625}(59,\cdot)\) \(\chi_{625}(64,\cdot)\) \(\chi_{625}(69,\cdot)\) \(\chi_{625}(79,\cdot)\) \(\chi_{625}(84,\cdot)\) \(\chi_{625}(89,\cdot)\) \(\chi_{625}(94,\cdot)\) \(\chi_{625}(104,\cdot)\) \(\chi_{625}(109,\cdot)\) \(\chi_{625}(114,\cdot)\) \(\chi_{625}(119,\cdot)\) \(\chi_{625}(129,\cdot)\) \(\chi_{625}(134,\cdot)\) \(\chi_{625}(139,\cdot)\) \(\chi_{625}(144,\cdot)\) \(\chi_{625}(154,\cdot)\) \(\chi_{625}(159,\cdot)\) \(\chi_{625}(164,\cdot)\) \(\chi_{625}(169,\cdot)\) \(\chi_{625}(179,\cdot)\) \(\chi_{625}(184,\cdot)\) ...

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

\(2\) → \(e\left(\frac{1}{250}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 625 }(4, a) \) \(1\)\(1\)\(e\left(\frac{1}{250}\right)\)\(e\left(\frac{107}{250}\right)\)\(e\left(\frac{1}{125}\right)\)\(e\left(\frac{54}{125}\right)\)\(e\left(\frac{47}{50}\right)\)\(e\left(\frac{3}{250}\right)\)\(e\left(\frac{107}{125}\right)\)\(e\left(\frac{113}{125}\right)\)\(e\left(\frac{109}{250}\right)\)\(e\left(\frac{139}{250}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 625 }(4,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 625 }(4,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 625 }(4,·),\chi_{ 625 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 625 }(4,·)) \;\) at \(\; a,b = \) e.g. 1,2