Properties

Label 125.6
Modulus $125$
Conductor $125$
Order $25$
Real no
Primitive yes
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(125, base_ring=CyclotomicField(50))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([4]))
 
pari: [g,chi] = znchar(Mod(6,125))
 

Basic properties

Modulus: \(125\)
Conductor: \(125\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(25\)
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 125.g

\(\chi_{125}(6,\cdot)\) \(\chi_{125}(11,\cdot)\) \(\chi_{125}(16,\cdot)\) \(\chi_{125}(21,\cdot)\) \(\chi_{125}(31,\cdot)\) \(\chi_{125}(36,\cdot)\) \(\chi_{125}(41,\cdot)\) \(\chi_{125}(46,\cdot)\) \(\chi_{125}(56,\cdot)\) \(\chi_{125}(61,\cdot)\) \(\chi_{125}(66,\cdot)\) \(\chi_{125}(71,\cdot)\) \(\chi_{125}(81,\cdot)\) \(\chi_{125}(86,\cdot)\) \(\chi_{125}(91,\cdot)\) \(\chi_{125}(96,\cdot)\) \(\chi_{125}(106,\cdot)\) \(\chi_{125}(111,\cdot)\) \(\chi_{125}(116,\cdot)\) \(\chi_{125}(121,\cdot)\)

sage: chi.galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{25})\)
Fixed field: 25.25.338813178901720135627329000271856784820556640625.1

Values on generators

\(2\) → \(e\left(\frac{2}{25}\right)\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 125 }(6, a) \) \(1\)\(1\)\(e\left(\frac{2}{25}\right)\)\(e\left(\frac{14}{25}\right)\)\(e\left(\frac{4}{25}\right)\)\(e\left(\frac{16}{25}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{6}{25}\right)\)\(e\left(\frac{3}{25}\right)\)\(e\left(\frac{2}{25}\right)\)\(e\left(\frac{18}{25}\right)\)\(e\left(\frac{3}{25}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 125 }(6,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 125 }(6,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 125 }(6,·),\chi_{ 125 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 125 }(6,·)) \;\) at \(\; a,b = \) e.g. 1,2