Properties

Label 125.4
Modulus $125$
Conductor $125$
Order $50$
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([1]))
 
pari: [g,chi] = znchar(Mod(4,125))
 

Basic properties

Modulus: \(125\)
Conductor: \(125\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(50\)
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.h

\(\chi_{125}(4,\cdot)\) \(\chi_{125}(9,\cdot)\) \(\chi_{125}(14,\cdot)\) \(\chi_{125}(19,\cdot)\) \(\chi_{125}(29,\cdot)\) \(\chi_{125}(34,\cdot)\) \(\chi_{125}(39,\cdot)\) \(\chi_{125}(44,\cdot)\) \(\chi_{125}(54,\cdot)\) \(\chi_{125}(59,\cdot)\) \(\chi_{125}(64,\cdot)\) \(\chi_{125}(69,\cdot)\) \(\chi_{125}(79,\cdot)\) \(\chi_{125}(84,\cdot)\) \(\chi_{125}(89,\cdot)\) \(\chi_{125}(94,\cdot)\) \(\chi_{125}(104,\cdot)\) \(\chi_{125}(109,\cdot)\) \(\chi_{125}(114,\cdot)\) \(\chi_{125}(119,\cdot)\)

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: Number field defined by a degree 50 polynomial

Values on generators

\(2\) → \(e\left(\frac{1}{50}\right)\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 125 }(4, a) \) \(1\)\(1\)\(e\left(\frac{1}{50}\right)\)\(e\left(\frac{7}{50}\right)\)\(e\left(\frac{1}{25}\right)\)\(e\left(\frac{4}{25}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{3}{50}\right)\)\(e\left(\frac{7}{25}\right)\)\(e\left(\frac{13}{25}\right)\)\(e\left(\frac{9}{50}\right)\)\(e\left(\frac{39}{50}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 125 }(4,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 125 }(4,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 125 }(4,·),\chi_{ 125 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 125 }(4,·)) \;\) at \(\; a,b = \) e.g. 1,2