Properties

Label 125.73
Modulus $125$
Conductor $125$
Order $100$
Real no
Primitive yes
Minimal yes
Parity odd

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Show commands: Pari/GP / SageMath
sage: from sage.modular.dirichlet import DirichletCharacter
 
sage: H = DirichletGroup(125, base_ring=CyclotomicField(100))
 
sage: M = H._module
 
sage: chi = DirichletCharacter(H, M([91]))
 
pari: [g,chi] = znchar(Mod(73,125))
 

Basic properties

Modulus: \(125\)
Conductor: \(125\)
sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Order: \(100\)
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Real: no
Primitive: yes
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 

Galois orbit 125.i

\(\chi_{125}(2,\cdot)\) \(\chi_{125}(3,\cdot)\) \(\chi_{125}(8,\cdot)\) \(\chi_{125}(12,\cdot)\) \(\chi_{125}(13,\cdot)\) \(\chi_{125}(17,\cdot)\) \(\chi_{125}(22,\cdot)\) \(\chi_{125}(23,\cdot)\) \(\chi_{125}(27,\cdot)\) \(\chi_{125}(28,\cdot)\) \(\chi_{125}(33,\cdot)\) \(\chi_{125}(37,\cdot)\) \(\chi_{125}(38,\cdot)\) \(\chi_{125}(42,\cdot)\) \(\chi_{125}(47,\cdot)\) \(\chi_{125}(48,\cdot)\) \(\chi_{125}(52,\cdot)\) \(\chi_{125}(53,\cdot)\) \(\chi_{125}(58,\cdot)\) \(\chi_{125}(62,\cdot)\) \(\chi_{125}(63,\cdot)\) \(\chi_{125}(67,\cdot)\) \(\chi_{125}(72,\cdot)\) \(\chi_{125}(73,\cdot)\) \(\chi_{125}(77,\cdot)\) \(\chi_{125}(78,\cdot)\) \(\chi_{125}(83,\cdot)\) \(\chi_{125}(87,\cdot)\) \(\chi_{125}(88,\cdot)\) \(\chi_{125}(92,\cdot)\) ...

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: $\Q(\zeta_{100})$
Fixed field: Number field defined by a degree 100 polynomial

Values on generators

\(2\) → \(e\left(\frac{91}{100}\right)\)

Values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 125 }(73, a) \) \(-1\)\(1\)\(e\left(\frac{91}{100}\right)\)\(e\left(\frac{37}{100}\right)\)\(e\left(\frac{41}{50}\right)\)\(e\left(\frac{7}{25}\right)\)\(e\left(\frac{7}{20}\right)\)\(e\left(\frac{73}{100}\right)\)\(e\left(\frac{37}{50}\right)\)\(e\left(\frac{4}{25}\right)\)\(e\left(\frac{19}{100}\right)\)\(e\left(\frac{49}{100}\right)\)
sage: chi.jacobi_sum(n)
 
\( \chi_{ 125 }(73,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

sage: chi.gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 125 }(73,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

sage: chi.jacobi_sum(n)
 
\( J(\chi_{ 125 }(73,·),\chi_{ 125 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

sage: chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 125 }(73,·)) \;\) at \(\; a,b = \) e.g. 1,2