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Properties

Label 8993.58
Modulus $8993$
Conductor $8993$
Order $4048$
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(8993, base_ring=CyclotomicField(4048)) M = H._module chi = DirichletCharacter(H, M([2783,2800]))
 
Copy content gp:[g,chi] = znchar(Mod(58, 8993))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("8993.58");
 

Basic properties

Modulus: \(8993\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(8993\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(4048\)
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 8993.bm

\(\chi_{8993}(3,\cdot)\) \(\chi_{8993}(6,\cdot)\) \(\chi_{8993}(12,\cdot)\) \(\chi_{8993}(27,\cdot)\) \(\chi_{8993}(29,\cdot)\) \(\chi_{8993}(31,\cdot)\) \(\chi_{8993}(39,\cdot)\) \(\chi_{8993}(41,\cdot)\) \(\chi_{8993}(48,\cdot)\) \(\chi_{8993}(54,\cdot)\) \(\chi_{8993}(58,\cdot)\) \(\chi_{8993}(62,\cdot)\) \(\chi_{8993}(71,\cdot)\) \(\chi_{8993}(73,\cdot)\) \(\chi_{8993}(75,\cdot)\) \(\chi_{8993}(78,\cdot)\) \(\chi_{8993}(82,\cdot)\) \(\chi_{8993}(95,\cdot)\) \(\chi_{8993}(96,\cdot)\) \(\chi_{8993}(105,\cdot)\) \(\chi_{8993}(108,\cdot)\) \(\chi_{8993}(124,\cdot)\) \(\chi_{8993}(131,\cdot)\) \(\chi_{8993}(133,\cdot)\) \(\chi_{8993}(141,\cdot)\) \(\chi_{8993}(142,\cdot)\) \(\chi_{8993}(146,\cdot)\) \(\chi_{8993}(147,\cdot)\) \(\chi_{8993}(150,\cdot)\) \(\chi_{8993}(156,\cdot)\) ...

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: $\Q(\zeta_{4048})$
Fixed field: Number field defined by a degree 4048 polynomial (not computed)

Values on generators

\((530,7940)\) → \((e\left(\frac{11}{16}\right),e\left(\frac{175}{253}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 8993 }(58, a) \) \(-1\)\(1\)\(e\left(\frac{1953}{2024}\right)\)\(e\left(\frac{3055}{4048}\right)\)\(e\left(\frac{941}{1012}\right)\)\(e\left(\frac{523}{4048}\right)\)\(e\left(\frac{2913}{4048}\right)\)\(e\left(\frac{3205}{4048}\right)\)\(e\left(\frac{1811}{2024}\right)\)\(e\left(\frac{1031}{2024}\right)\)\(e\left(\frac{381}{4048}\right)\)\(e\left(\frac{1561}{4048}\right)\)
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)
 
\( \chi_{ 8993 }(58,a) \;\) at \(\;a = \) e.g. 2