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Properties

Label 763.102
Modulus $763$
Conductor $763$
Order $54$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(763, base_ring=CyclotomicField(54)) M = H._module chi = DirichletCharacter(H, M([36,47]))
 
Copy content pari:[g,chi] = znchar(Mod(102,763))
 

Basic properties

Modulus: \(763\)
Conductor: \(763\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(54\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: yes
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 763.cf

\(\chi_{763}(100,\cdot)\) \(\chi_{763}(102,\cdot)\) \(\chi_{763}(137,\cdot)\) \(\chi_{763}(254,\cdot)\) \(\chi_{763}(347,\cdot)\) \(\chi_{763}(387,\cdot)\) \(\chi_{763}(401,\cdot)\) \(\chi_{763}(410,\cdot)\) \(\chi_{763}(415,\cdot)\) \(\chi_{763}(431,\cdot)\) \(\chi_{763}(520,\cdot)\) \(\chi_{763}(557,\cdot)\) \(\chi_{763}(576,\cdot)\) \(\chi_{763}(606,\cdot)\) \(\chi_{763}(632,\cdot)\) \(\chi_{763}(639,\cdot)\) \(\chi_{763}(683,\cdot)\) \(\chi_{763}(760,\cdot)\)

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: \(\Q(\zeta_{27})\)
Fixed field: Number field defined by a degree 54 polynomial

Values on generators

\((437,442)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{47}{54}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 763 }(102, a) \) \(1\)\(1\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{25}{27}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{13}{27}\right)\)\(e\left(\frac{47}{54}\right)\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{23}{27}\right)\)\(e\left(\frac{23}{54}\right)\)\(e\left(\frac{49}{54}\right)\)\(e\left(\frac{22}{27}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 763 }(102,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 763 }(102,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 763 }(102,·),\chi_{ 763 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 763 }(102,·)) \;\) at \(\; a,b = \) e.g. 1,2