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Properties

Label 413.83
Modulus $413$
Conductor $413$
Order $58$
Real no
Primitive yes
Minimal yes
Parity even

Related objects

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Show commands: Magma / Pari/GP / SageMath
Copy content comment:Define the Dirichlet character
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(413, base_ring=CyclotomicField(58)) M = H._module chi = DirichletCharacter(H, M([29,53]))
 
Copy content gp:[g,chi] = znchar(Mod(83, 413))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("413.83");
 

Basic properties

Modulus: \(413\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(413\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(58\)
Copy content comment:Order
 
Copy content sage:chi.multiplicative_order()
 
Copy content gp:charorder(g,chi)
 
Copy content magma:Order(chi);
 
Real: no
Copy content comment:Whether the character is real
 
Copy content sage:chi.multiplicative_order() <= 2
 
Copy content gp:charorder(g,chi) <= 2
 
Copy content magma:Order(chi) le 2;
 
Primitive: yes
Copy content comment:If the character is primitive
 
Copy content sage:chi.is_primitive()
 
Copy content gp:#znconreyconductor(g,chi)==1
 
Copy content magma:IsPrimitive(chi);
 
Minimal: yes
Parity: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 413.l

\(\chi_{413}(6,\cdot)\) \(\chi_{413}(13,\cdot)\) \(\chi_{413}(34,\cdot)\) \(\chi_{413}(55,\cdot)\) \(\chi_{413}(69,\cdot)\) \(\chi_{413}(83,\cdot)\) \(\chi_{413}(90,\cdot)\) \(\chi_{413}(97,\cdot)\) \(\chi_{413}(111,\cdot)\) \(\chi_{413}(132,\cdot)\) \(\chi_{413}(160,\cdot)\) \(\chi_{413}(174,\cdot)\) \(\chi_{413}(188,\cdot)\) \(\chi_{413}(195,\cdot)\) \(\chi_{413}(209,\cdot)\) \(\chi_{413}(216,\cdot)\) \(\chi_{413}(244,\cdot)\) \(\chi_{413}(279,\cdot)\) \(\chi_{413}(286,\cdot)\) \(\chi_{413}(328,\cdot)\) \(\chi_{413}(335,\cdot)\) \(\chi_{413}(342,\cdot)\) \(\chi_{413}(349,\cdot)\) \(\chi_{413}(356,\cdot)\) \(\chi_{413}(377,\cdot)\) \(\chi_{413}(384,\cdot)\) \(\chi_{413}(391,\cdot)\) \(\chi_{413}(398,\cdot)\)

Copy content comment:Galois orbit
 
Copy content sage:chi.galois_orbit()
 
Copy content gp:order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 
Copy content magma:order := Order(chi); { chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
 

Related number fields

Field of values: $\Q(\zeta_{29})$
Fixed field: Number field defined by a degree 58 polynomial

Values on generators

\((178,120)\) → \((-1,e\left(\frac{53}{58}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 413 }(83, a) \) \(1\)\(1\)\(e\left(\frac{53}{58}\right)\)\(e\left(\frac{11}{58}\right)\)\(e\left(\frac{24}{29}\right)\)\(e\left(\frac{57}{58}\right)\)\(e\left(\frac{3}{29}\right)\)\(e\left(\frac{43}{58}\right)\)\(e\left(\frac{11}{29}\right)\)\(e\left(\frac{26}{29}\right)\)\(e\left(\frac{49}{58}\right)\)\(e\left(\frac{1}{58}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x)
 
Copy content gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
 
Copy content magma:chi(x)
 
\( \chi_{ 413 }(83,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content comment:Gauss sum
 
Copy content sage:chi.gauss_sum(a)
 
Copy content gp:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 413 }(83,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content comment:Jacobi sum
 
Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 413 }(83,·),\chi_{ 413 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content comment:Kloosterman sum
 
Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 413 }(83,·)) \;\) at \(\; a,b = \) e.g. 1,2