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Properties

Label 4307.39
Modulus $4307$
Conductor $4307$
Order $2088$
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(4307, base_ring=CyclotomicField(2088)) M = H._module chi = DirichletCharacter(H, M([1332,1885]))
 
Copy content gp:[g,chi] = znchar(Mod(39, 4307))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4307.39");
 

Basic properties

Modulus: \(4307\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4307\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(2088\)
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 4307.bu

\(\chi_{4307}(11,\cdot)\) \(\chi_{4307}(13,\cdot)\) \(\chi_{4307}(14,\cdot)\) \(\chi_{4307}(31,\cdot)\) \(\chi_{4307}(33,\cdot)\) \(\chi_{4307}(34,\cdot)\) \(\chi_{4307}(39,\cdot)\) \(\chi_{4307}(40,\cdot)\) \(\chi_{4307}(42,\cdot)\) \(\chi_{4307}(44,\cdot)\) \(\chi_{4307}(47,\cdot)\) \(\chi_{4307}(93,\cdot)\) \(\chi_{4307}(99,\cdot)\) \(\chi_{4307}(101,\cdot)\) \(\chi_{4307}(102,\cdot)\) \(\chi_{4307}(106,\cdot)\) \(\chi_{4307}(113,\cdot)\) \(\chi_{4307}(115,\cdot)\) \(\chi_{4307}(120,\cdot)\) \(\chi_{4307}(126,\cdot)\) \(\chi_{4307}(131,\cdot)\) \(\chi_{4307}(132,\cdot)\) \(\chi_{4307}(141,\cdot)\) \(\chi_{4307}(151,\cdot)\) \(\chi_{4307}(157,\cdot)\) \(\chi_{4307}(160,\cdot)\) \(\chi_{4307}(161,\cdot)\) \(\chi_{4307}(172,\cdot)\) \(\chi_{4307}(174,\cdot)\) \(\chi_{4307}(179,\cdot)\) ...

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: $\Q(\zeta_{2088})$
Fixed field: Number field defined by a degree 2088 polynomial (not computed)

Values on generators

\((2775,1830)\) → \((e\left(\frac{37}{58}\right),e\left(\frac{65}{72}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 4307 }(39, a) \) \(1\)\(1\)\(e\left(\frac{449}{522}\right)\)\(e\left(\frac{109}{348}\right)\)\(e\left(\frac{188}{261}\right)\)\(e\left(\frac{1525}{2088}\right)\)\(e\left(\frac{181}{1044}\right)\)\(e\left(\frac{191}{696}\right)\)\(e\left(\frac{101}{174}\right)\)\(e\left(\frac{109}{174}\right)\)\(e\left(\frac{137}{232}\right)\)\(e\left(\frac{1255}{2088}\right)\)
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)
 
\( \chi_{ 4307 }(39,a) \;\) at \(\;a = \) e.g. 2