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Properties

Label 1777.137
Modulus $1777$
Conductor $1777$
Order $888$
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(1777, base_ring=CyclotomicField(888)) M = H._module chi = DirichletCharacter(H, M([319]))
 
Copy content gp:[g,chi] = znchar(Mod(137, 1777))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1777.137");
 

Basic properties

Modulus: \(1777\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1777\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(888\)
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 1777.s

\(\chi_{1777}(3,\cdot)\) \(\chi_{1777}(6,\cdot)\) \(\chi_{1777}(12,\cdot)\) \(\chi_{1777}(13,\cdot)\) \(\chi_{1777}(24,\cdot)\) \(\chi_{1777}(25,\cdot)\) \(\chi_{1777}(26,\cdot)\) \(\chi_{1777}(31,\cdot)\) \(\chi_{1777}(37,\cdot)\) \(\chi_{1777}(43,\cdot)\) \(\chi_{1777}(48,\cdot)\) \(\chi_{1777}(50,\cdot)\) \(\chi_{1777}(52,\cdot)\) \(\chi_{1777}(62,\cdot)\) \(\chi_{1777}(71,\cdot)\) \(\chi_{1777}(74,\cdot)\) \(\chi_{1777}(86,\cdot)\) \(\chi_{1777}(95,\cdot)\) \(\chi_{1777}(96,\cdot)\) \(\chi_{1777}(100,\cdot)\) \(\chi_{1777}(104,\cdot)\) \(\chi_{1777}(113,\cdot)\) \(\chi_{1777}(117,\cdot)\) \(\chi_{1777}(124,\cdot)\) \(\chi_{1777}(137,\cdot)\) \(\chi_{1777}(142,\cdot)\) \(\chi_{1777}(148,\cdot)\) \(\chi_{1777}(151,\cdot)\) \(\chi_{1777}(153,\cdot)\) \(\chi_{1777}(159,\cdot)\) ...

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: $\Q(\zeta_{888})$
Fixed field: Number field defined by a degree 888 polynomial (not computed)

Values on generators

\(5\) → \(e\left(\frac{319}{888}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1777 }(137, a) \) \(1\)\(1\)\(e\left(\frac{17}{37}\right)\)\(e\left(\frac{271}{444}\right)\)\(e\left(\frac{34}{37}\right)\)\(e\left(\frac{319}{888}\right)\)\(e\left(\frac{31}{444}\right)\)\(e\left(\frac{87}{296}\right)\)\(e\left(\frac{14}{37}\right)\)\(e\left(\frac{49}{222}\right)\)\(e\left(\frac{727}{888}\right)\)\(e\left(\frac{11}{24}\right)\)
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)
 
\( \chi_{ 1777 }(137,a) \;\) at \(\;a = \) e.g. 2