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Properties

Label 14641.7
Modulus $14641$
Conductor $14641$
Order $13310$
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(14641, base_ring=CyclotomicField(13310)) M = H._module chi = DirichletCharacter(H, M([2977]))
 
Copy content gp:[g,chi] = znchar(Mod(7, 14641))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("14641.7");
 

Basic properties

Modulus: \(14641\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(14641\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(13310\)
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 14641.p

\(\chi_{14641}(2,\cdot)\) \(\chi_{14641}(6,\cdot)\) \(\chi_{14641}(7,\cdot)\) \(\chi_{14641}(8,\cdot)\) \(\chi_{14641}(13,\cdot)\) \(\chi_{14641}(17,\cdot)\) \(\chi_{14641}(18,\cdot)\) \(\chi_{14641}(19,\cdot)\) \(\chi_{14641}(24,\cdot)\) \(\chi_{14641}(28,\cdot)\) \(\chi_{14641}(29,\cdot)\) \(\chi_{14641}(30,\cdot)\) \(\chi_{14641}(35,\cdot)\) \(\chi_{14641}(39,\cdot)\) \(\chi_{14641}(41,\cdot)\) \(\chi_{14641}(46,\cdot)\) \(\chi_{14641}(50,\cdot)\) \(\chi_{14641}(51,\cdot)\) \(\chi_{14641}(52,\cdot)\) \(\chi_{14641}(57,\cdot)\) \(\chi_{14641}(61,\cdot)\) \(\chi_{14641}(62,\cdot)\) \(\chi_{14641}(63,\cdot)\) \(\chi_{14641}(68,\cdot)\) \(\chi_{14641}(72,\cdot)\) \(\chi_{14641}(73,\cdot)\) \(\chi_{14641}(74,\cdot)\) \(\chi_{14641}(79,\cdot)\) \(\chi_{14641}(83,\cdot)\) \(\chi_{14641}(84,\cdot)\) ...

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: $\Q(\zeta_{6655})$
Fixed field: Number field defined by a degree 13310 polynomial (not computed)

Values on generators

\(2\) → \(e\left(\frac{2977}{13310}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 14641 }(7, a) \) \(-1\)\(1\)\(e\left(\frac{2977}{13310}\right)\)\(e\left(\frac{478}{605}\right)\)\(e\left(\frac{2977}{6655}\right)\)\(e\left(\frac{424}{6655}\right)\)\(e\left(\frac{183}{13310}\right)\)\(e\left(\frac{11379}{13310}\right)\)\(e\left(\frac{8931}{13310}\right)\)\(e\left(\frac{351}{605}\right)\)\(e\left(\frac{765}{2662}\right)\)\(e\left(\frac{316}{1331}\right)\)
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)
 
\( \chi_{ 14641 }(7,a) \;\) at \(\;a = \) e.g. 2