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Properties

Label 8077.53
Modulus $8077$
Conductor $8077$
Order $1960$
Real no
Primitive yes
Minimal yes
Parity odd

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(8077, base_ring=CyclotomicField(1960)) M = H._module chi = DirichletCharacter(H, M([1323,720]))
 
Copy content gp:[g,chi] = znchar(Mod(53, 8077))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8077.53");
 

Basic properties

Modulus: \(8077\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(8077\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1960\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 8077.dd

\(\chi_{8077}(24,\cdot)\) \(\chi_{8077}(28,\cdot)\) \(\chi_{8077}(29,\cdot)\) \(\chi_{8077}(34,\cdot)\) \(\chi_{8077}(53,\cdot)\) \(\chi_{8077}(54,\cdot)\) \(\chi_{8077}(60,\cdot)\) \(\chi_{8077}(63,\cdot)\) \(\chi_{8077}(70,\cdot)\) \(\chi_{8077}(76,\cdot)\) \(\chi_{8077}(88,\cdot)\) \(\chi_{8077}(101,\cdot)\) \(\chi_{8077}(135,\cdot)\) \(\chi_{8077}(142,\cdot)\) \(\chi_{8077}(158,\cdot)\) \(\chi_{8077}(171,\cdot)\) \(\chi_{8077}(175,\cdot)\) \(\chi_{8077}(188,\cdot)\) \(\chi_{8077}(190,\cdot)\) \(\chi_{8077}(193,\cdot)\) \(\chi_{8077}(220,\cdot)\) \(\chi_{8077}(231,\cdot)\) \(\chi_{8077}(234,\cdot)\) \(\chi_{8077}(239,\cdot)\) \(\chi_{8077}(257,\cdot)\) \(\chi_{8077}(258,\cdot)\) \(\chi_{8077}(298,\cdot)\) \(\chi_{8077}(302,\cdot)\) \(\chi_{8077}(339,\cdot)\) \(\chi_{8077}(347,\cdot)\) ...

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: $\Q(\zeta_{1960})$
Fixed field: Number field defined by a degree 1960 polynomial (not computed)

Values on generators

\((4926,7094)\) → \((e\left(\frac{27}{40}\right),e\left(\frac{18}{49}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 8077 }(53, a) \) \(-1\)\(1\)\(e\left(\frac{899}{980}\right)\)\(e\left(\frac{241}{392}\right)\)\(e\left(\frac{409}{490}\right)\)\(e\left(\frac{533}{980}\right)\)\(e\left(\frac{149}{280}\right)\)\(e\left(\frac{1877}{1960}\right)\)\(e\left(\frac{737}{980}\right)\)\(e\left(\frac{45}{196}\right)\)\(e\left(\frac{113}{245}\right)\)\(e\left(\frac{1329}{1960}\right)\)
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)
 
\( \chi_{ 8077 }(53,a) \;\) at \(\;a = \) e.g. 2