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Properties

Label 553.494
Modulus $553$
Conductor $553$
Order $39$
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(553, base_ring=CyclotomicField(78)) M = H._module chi = DirichletCharacter(H, M([52,70]))
 
Copy content pari:[g,chi] = znchar(Mod(494,553))
 

Basic properties

Modulus: \(553\)
Conductor: \(553\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(39\)
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 553.y

\(\chi_{553}(9,\cdot)\) \(\chi_{553}(25,\cdot)\) \(\chi_{553}(72,\cdot)\) \(\chi_{553}(81,\cdot)\) \(\chi_{553}(95,\cdot)\) \(\chi_{553}(123,\cdot)\) \(\chi_{553}(130,\cdot)\) \(\chi_{553}(163,\cdot)\) \(\chi_{553}(177,\cdot)\) \(\chi_{553}(184,\cdot)\) \(\chi_{553}(198,\cdot)\) \(\chi_{553}(200,\cdot)\) \(\chi_{553}(207,\cdot)\) \(\chi_{553}(268,\cdot)\) \(\chi_{553}(310,\cdot)\) \(\chi_{553}(361,\cdot)\) \(\chi_{553}(366,\cdot)\) \(\chi_{553}(431,\cdot)\) \(\chi_{553}(478,\cdot)\) \(\chi_{553}(485,\cdot)\) \(\chi_{553}(487,\cdot)\) \(\chi_{553}(494,\cdot)\) \(\chi_{553}(506,\cdot)\) \(\chi_{553}(550,\cdot)\)

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: $\Q(\zeta_{39})$
Fixed field: 39.39.12088669642594427834079815641631684897704150233955220736437592661471356964310045748175158745489.2

Values on generators

\((80,477)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{35}{39}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 553 }(494, a) \) \(1\)\(1\)\(e\left(\frac{12}{13}\right)\)\(e\left(\frac{22}{39}\right)\)\(e\left(\frac{11}{13}\right)\)\(e\left(\frac{38}{39}\right)\)\(e\left(\frac{19}{39}\right)\)\(e\left(\frac{10}{13}\right)\)\(e\left(\frac{5}{39}\right)\)\(e\left(\frac{35}{39}\right)\)\(e\left(\frac{9}{13}\right)\)\(e\left(\frac{16}{39}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 553 }(494,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 553 }(494,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 553 }(494,·),\chi_{ 553 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 553 }(494,·)) \;\) at \(\; a,b = \) e.g. 1,2