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Properties

Label 96800.13117
Modulus $96800$
Conductor $96800$
Order $440$
Real no
Primitive yes
Minimal yes
Parity odd

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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(96800, base_ring=CyclotomicField(440)) M = H._module chi = DirichletCharacter(H, M([0,165,286,56]))
 
Copy content gp:[g,chi] = znchar(Mod(13117, 96800))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("96800.13117");
 

Basic properties

Modulus: \(96800\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(96800\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(440\)
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 96800.bdv

\(\chi_{96800}(53,\cdot)\) \(\chi_{96800}(213,\cdot)\) \(\chi_{96800}(533,\cdot)\) \(\chi_{96800}(773,\cdot)\) \(\chi_{96800}(1197,\cdot)\) \(\chi_{96800}(2237,\cdot)\) \(\chi_{96800}(3677,\cdot)\) \(\chi_{96800}(4317,\cdot)\) \(\chi_{96800}(4453,\cdot)\) \(\chi_{96800}(4613,\cdot)\) \(\chi_{96800}(4933,\cdot)\) \(\chi_{96800}(5173,\cdot)\) \(\chi_{96800}(5597,\cdot)\) \(\chi_{96800}(6637,\cdot)\) \(\chi_{96800}(8077,\cdot)\) \(\chi_{96800}(8717,\cdot)\) \(\chi_{96800}(8853,\cdot)\) \(\chi_{96800}(9013,\cdot)\) \(\chi_{96800}(9333,\cdot)\) \(\chi_{96800}(9573,\cdot)\) \(\chi_{96800}(9997,\cdot)\) \(\chi_{96800}(11037,\cdot)\) \(\chi_{96800}(12477,\cdot)\) \(\chi_{96800}(13117,\cdot)\) \(\chi_{96800}(13253,\cdot)\) \(\chi_{96800}(13413,\cdot)\) \(\chi_{96800}(13733,\cdot)\) \(\chi_{96800}(13973,\cdot)\) \(\chi_{96800}(14397,\cdot)\) \(\chi_{96800}(15437,\cdot)\) ...

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: $\Q(\zeta_{440})$
Fixed field: Number field defined by a degree 440 polynomial (not computed)

Values on generators

\((90751,12101,30977,14401)\) → \((1,e\left(\frac{3}{8}\right),e\left(\frac{13}{20}\right),e\left(\frac{7}{55}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 96800 }(13117, a) \) \(-1\)\(1\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{49}{55}\right)\)\(-i\)\(e\left(\frac{73}{88}\right)\)\(e\left(\frac{41}{220}\right)\)\(e\left(\frac{391}{440}\right)\)\(e\left(\frac{337}{440}\right)\)\(e\left(\frac{17}{55}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{259}{440}\right)\)
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)
 
\( \chi_{ 96800 }(13117,a) \;\) at \(\;a = \) e.g. 2