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Properties

Label 2399.44
Modulus $2399$
Conductor $2399$
Order $2398$
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(2399, base_ring=CyclotomicField(2398)) M = H._module chi = DirichletCharacter(H, M([323]))
 
Copy content gp:[g,chi] = znchar(Mod(44, 2399))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2399.44");
 

Basic properties

Modulus: \(2399\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(2399\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(2398\)
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 2399.h

\(\chi_{2399}(11,\cdot)\) \(\chi_{2399}(13,\cdot)\) \(\chi_{2399}(19,\cdot)\) \(\chi_{2399}(22,\cdot)\) \(\chi_{2399}(26,\cdot)\) \(\chi_{2399}(29,\cdot)\) \(\chi_{2399}(33,\cdot)\) \(\chi_{2399}(37,\cdot)\) \(\chi_{2399}(38,\cdot)\) \(\chi_{2399}(39,\cdot)\) \(\chi_{2399}(44,\cdot)\) \(\chi_{2399}(52,\cdot)\) \(\chi_{2399}(53,\cdot)\) \(\chi_{2399}(55,\cdot)\) \(\chi_{2399}(57,\cdot)\) \(\chi_{2399}(58,\cdot)\) \(\chi_{2399}(65,\cdot)\) \(\chi_{2399}(66,\cdot)\) \(\chi_{2399}(67,\cdot)\) \(\chi_{2399}(74,\cdot)\) \(\chi_{2399}(76,\cdot)\) \(\chi_{2399}(78,\cdot)\) \(\chi_{2399}(83,\cdot)\) \(\chi_{2399}(87,\cdot)\) \(\chi_{2399}(94,\cdot)\) \(\chi_{2399}(95,\cdot)\) \(\chi_{2399}(97,\cdot)\) \(\chi_{2399}(99,\cdot)\) \(\chi_{2399}(103,\cdot)\) \(\chi_{2399}(106,\cdot)\) ...

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: $\Q(\zeta_{1199})$
Fixed field: Number field defined by a degree 2398 polynomial (not computed)

Values on generators

\(11\) → \(e\left(\frac{323}{2398}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 2399 }(44, a) \) \(-1\)\(1\)\(e\left(\frac{223}{1199}\right)\)\(e\left(\frac{1103}{1199}\right)\)\(e\left(\frac{446}{1199}\right)\)\(e\left(\frac{90}{1199}\right)\)\(e\left(\frac{127}{1199}\right)\)\(e\left(\frac{955}{1199}\right)\)\(e\left(\frac{669}{1199}\right)\)\(e\left(\frac{1007}{1199}\right)\)\(e\left(\frac{313}{1199}\right)\)\(e\left(\frac{323}{2398}\right)\)
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)
 
\( \chi_{ 2399 }(44,a) \;\) at \(\;a = \) e.g. 2