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Properties

Label 1311.749
Modulus $1311$
Conductor $1311$
Order $66$
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(1311, base_ring=CyclotomicField(66)) M = H._module chi = DirichletCharacter(H, M([33,11,42]))
 
Copy content gp:[g,chi] = znchar(Mod(749, 1311))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1311.749");
 

Basic properties

Modulus: \(1311\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1311\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(66\)
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 1311.bl

\(\chi_{1311}(8,\cdot)\) \(\chi_{1311}(50,\cdot)\) \(\chi_{1311}(164,\cdot)\) \(\chi_{1311}(179,\cdot)\) \(\chi_{1311}(236,\cdot)\) \(\chi_{1311}(278,\cdot)\) \(\chi_{1311}(335,\cdot)\) \(\chi_{1311}(407,\cdot)\) \(\chi_{1311}(449,\cdot)\) \(\chi_{1311}(464,\cdot)\) \(\chi_{1311}(578,\cdot)\) \(\chi_{1311}(692,\cdot)\) \(\chi_{1311}(749,\cdot)\) \(\chi_{1311}(791,\cdot)\) \(\chi_{1311}(863,\cdot)\) \(\chi_{1311}(905,\cdot)\) \(\chi_{1311}(1076,\cdot)\) \(\chi_{1311}(1133,\cdot)\) \(\chi_{1311}(1205,\cdot)\) \(\chi_{1311}(1304,\cdot)\)

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: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 66 polynomial

Values on generators

\((875,553,856)\) → \((-1,e\left(\frac{1}{6}\right),e\left(\frac{7}{11}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 1311 }(749, a) \) \(1\)\(1\)\(e\left(\frac{31}{33}\right)\)\(e\left(\frac{29}{33}\right)\)\(e\left(\frac{53}{66}\right)\)\(e\left(\frac{1}{11}\right)\)\(e\left(\frac{9}{11}\right)\)\(e\left(\frac{49}{66}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{49}{66}\right)\)\(e\left(\frac{1}{33}\right)\)\(e\left(\frac{25}{33}\right)\)
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)
 
\( \chi_{ 1311 }(749,a) \;\) at \(\;a = \) e.g. 2