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Properties

Label 3823.115
Modulus $3823$
Conductor $3823$
Order $3822$
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(3823, base_ring=CyclotomicField(3822)) M = H._module chi = DirichletCharacter(H, M([2543]))
 
Copy content gp:[g,chi] = znchar(Mod(115, 3823))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3823.115");
 

Basic properties

Modulus: \(3823\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(3823\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(3822\)
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 3823.x

\(\chi_{3823}(3,\cdot)\) \(\chi_{3823}(6,\cdot)\) \(\chi_{3823}(19,\cdot)\) \(\chi_{3823}(24,\cdot)\) \(\chi_{3823}(31,\cdot)\) \(\chi_{3823}(38,\cdot)\) \(\chi_{3823}(45,\cdot)\) \(\chi_{3823}(47,\cdot)\) \(\chi_{3823}(48,\cdot)\) \(\chi_{3823}(51,\cdot)\) \(\chi_{3823}(62,\cdot)\) \(\chi_{3823}(63,\cdot)\) \(\chi_{3823}(66,\cdot)\) \(\chi_{3823}(75,\cdot)\) \(\chi_{3823}(76,\cdot)\) \(\chi_{3823}(85,\cdot)\) \(\chi_{3823}(90,\cdot)\) \(\chi_{3823}(94,\cdot)\) \(\chi_{3823}(96,\cdot)\) \(\chi_{3823}(101,\cdot)\) \(\chi_{3823}(105,\cdot)\) \(\chi_{3823}(113,\cdot)\) \(\chi_{3823}(115,\cdot)\) \(\chi_{3823}(119,\cdot)\) \(\chi_{3823}(124,\cdot)\) \(\chi_{3823}(126,\cdot)\) \(\chi_{3823}(127,\cdot)\) \(\chi_{3823}(129,\cdot)\) \(\chi_{3823}(130,\cdot)\) \(\chi_{3823}(137,\cdot)\) ...

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: $\Q(\zeta_{1911})$
Fixed field: Number field defined by a degree 3822 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{2543}{3822}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 3823 }(115, a) \) \(-1\)\(1\)\(e\left(\frac{449}{637}\right)\)\(e\left(\frac{2543}{3822}\right)\)\(e\left(\frac{261}{637}\right)\)\(e\left(\frac{879}{1274}\right)\)\(e\left(\frac{1415}{3822}\right)\)\(e\left(\frac{33}{98}\right)\)\(e\left(\frac{73}{637}\right)\)\(e\left(\frac{632}{1911}\right)\)\(e\left(\frac{503}{1274}\right)\)\(e\left(\frac{366}{637}\right)\)
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)
 
\( \chi_{ 3823 }(115,a) \;\) at \(\;a = \) e.g. 2