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Properties

Label 459.19
Modulus $459$
Conductor $153$
Order $24$
Real no
Primitive no
Minimal no
Parity even

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(459, base_ring=CyclotomicField(24)) M = H._module chi = DirichletCharacter(H, M([16,21]))
 
Copy content pari:[g,chi] = znchar(Mod(19,459))
 

Basic properties

Modulus: \(459\)
Conductor: \(153\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(24\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{153}(70,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 459.v

\(\chi_{459}(19,\cdot)\) \(\chi_{459}(100,\cdot)\) \(\chi_{459}(127,\cdot)\) \(\chi_{459}(145,\cdot)\) \(\chi_{459}(172,\cdot)\) \(\chi_{459}(253,\cdot)\) \(\chi_{459}(280,\cdot)\) \(\chi_{459}(451,\cdot)\)

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

Related number fields

Field of values: \(\Q(\zeta_{24})\)
Fixed field: 24.24.128028748427622359924863503266793533356497.1

Values on generators

\((137,190)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{7}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 459 }(19, a) \) \(1\)\(1\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{17}{24}\right)\)\(e\left(\frac{7}{24}\right)\)\(-i\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{19}{24}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{5}{24}\right)\)\(e\left(\frac{2}{3}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 459 }(19,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 459 }(19,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 459 }(19,·),\chi_{ 459 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 459 }(19,·)) \;\) at \(\; a,b = \) e.g. 1,2