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Properties

Label 28767.797
Modulus $28767$
Conductor $28767$
Order $1554$
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(28767, base_ring=CyclotomicField(1554)) M = H._module chi = DirichletCharacter(H, M([777,592,1050]))
 
Copy content gp:[g,chi] = znchar(Mod(797, 28767))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("28767.797");
 

Basic properties

Modulus: \(28767\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(28767\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1554\)
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 28767.fn

\(\chi_{28767}(14,\cdot)\) \(\chi_{28767}(17,\cdot)\) \(\chi_{28767}(56,\cdot)\) \(\chi_{28767}(68,\cdot)\) \(\chi_{28767}(197,\cdot)\) \(\chi_{28767}(230,\cdot)\) \(\chi_{28767}(239,\cdot)\) \(\chi_{28767}(272,\cdot)\) \(\chi_{28767}(359,\cdot)\) \(\chi_{28767}(461,\cdot)\) \(\chi_{28767}(617,\cdot)\) \(\chi_{28767}(683,\cdot)\) \(\chi_{28767}(701,\cdot)\) \(\chi_{28767}(788,\cdot)\) \(\chi_{28767}(797,\cdot)\) \(\chi_{28767}(920,\cdot)\) \(\chi_{28767}(926,\cdot)\) \(\chi_{28767}(941,\cdot)\) \(\chi_{28767}(956,\cdot)\) \(\chi_{28767}(1004,\cdot)\) \(\chi_{28767}(1088,\cdot)\) \(\chi_{28767}(1175,\cdot)\) \(\chi_{28767}(1235,\cdot)\) \(\chi_{28767}(1346,\cdot)\) \(\chi_{28767}(1436,\cdot)\) \(\chi_{28767}(1457,\cdot)\) \(\chi_{28767}(1502,\cdot)\)

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: $\Q(\zeta_{777})$
Fixed field: Number field defined by a degree 1554 polynomial (not computed)

Values on generators

\((9590,12043,26317)\) → \((-1,e\left(\frac{8}{21}\right),e\left(\frac{25}{37}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 28767 }(797, a) \) \(-1\)\(1\)\(e\left(\frac{211}{518}\right)\)\(e\left(\frac{211}{259}\right)\)\(e\left(\frac{247}{1554}\right)\)\(e\left(\frac{25}{111}\right)\)\(e\left(\frac{115}{518}\right)\)\(e\left(\frac{440}{777}\right)\)\(e\left(\frac{117}{518}\right)\)\(e\left(\frac{400}{777}\right)\)\(e\left(\frac{983}{1554}\right)\)\(e\left(\frac{163}{259}\right)\)
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)
 
\( \chi_{ 28767 }(797,a) \;\) at \(\;a = \) e.g. 2