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Properties

Label 451.403
Modulus $451$
Conductor $451$
Order $40$
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(451, base_ring=CyclotomicField(40)) M = H._module chi = DirichletCharacter(H, M([28,19]))
 
Copy content pari:[g,chi] = znchar(Mod(403,451))
 

Basic properties

Modulus: \(451\)
Conductor: \(451\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(40\)
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 451.bz

\(\chi_{451}(7,\cdot)\) \(\chi_{451}(17,\cdot)\) \(\chi_{451}(129,\cdot)\) \(\chi_{451}(151,\cdot)\) \(\chi_{451}(183,\cdot)\) \(\chi_{451}(227,\cdot)\) \(\chi_{451}(270,\cdot)\) \(\chi_{451}(299,\cdot)\) \(\chi_{451}(343,\cdot)\) \(\chi_{451}(354,\cdot)\) \(\chi_{451}(358,\cdot)\) \(\chi_{451}(380,\cdot)\) \(\chi_{451}(382,\cdot)\) \(\chi_{451}(398,\cdot)\) \(\chi_{451}(403,\cdot)\) \(\chi_{451}(404,\cdot)\)

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: \(\Q(\zeta_{40})\)
Fixed field: 40.40.24474119586015991540399878761458343038838464370944248895385985016435908406965653861955404428151914121.3

Values on generators

\((288,375)\) → \((e\left(\frac{7}{10}\right),e\left(\frac{19}{40}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 451 }(403, a) \) \(1\)\(1\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{29}{40}\right)\)\(e\left(\frac{1}{10}\right)\)\(i\)\(e\left(\frac{31}{40}\right)\)\(e\left(\frac{17}{40}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{9}{20}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{33}{40}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 451 }(403,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 451 }(403,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 451 }(403,·),\chi_{ 451 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 451 }(403,·)) \;\) at \(\; a,b = \) e.g. 1,2