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Properties

Label 3723.1067
Modulus $3723$
Conductor $3723$
Order $72$
Real no
Primitive yes
Minimal yes
Parity even

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(3723, base_ring=CyclotomicField(72)) M = H._module chi = DirichletCharacter(H, M([36,18,13]))
 
Copy content gp:[g,chi] = znchar(Mod(1067, 3723))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("3723.1067");
 

Basic properties

Modulus: \(3723\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(3723\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(72\)
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: even
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 3723.fp

\(\chi_{3723}(404,\cdot)\) \(\chi_{3723}(506,\cdot)\) \(\chi_{3723}(599,\cdot)\) \(\chi_{3723}(701,\cdot)\) \(\chi_{3723}(710,\cdot)\) \(\chi_{3723}(905,\cdot)\) \(\chi_{3723}(1007,\cdot)\) \(\chi_{3723}(1067,\cdot)\) \(\chi_{3723}(1373,\cdot)\) \(\chi_{3723}(1619,\cdot)\) \(\chi_{3723}(1721,\cdot)\) \(\chi_{3723}(1985,\cdot)\) \(\chi_{3723}(2291,\cdot)\) \(\chi_{3723}(2435,\cdot)\) \(\chi_{3723}(2639,\cdot)\) \(\chi_{3723}(2648,\cdot)\) \(\chi_{3723}(2690,\cdot)\) \(\chi_{3723}(2852,\cdot)\) \(\chi_{3723}(2894,\cdot)\) \(\chi_{3723}(2954,\cdot)\) \(\chi_{3723}(3464,\cdot)\) \(\chi_{3723}(3608,\cdot)\) \(\chi_{3723}(3617,\cdot)\) \(\chi_{3723}(3710,\cdot)\)

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: $\Q(\zeta_{72})$
Fixed field: Number field defined by a degree 72 polynomial

Values on generators

\((2483,3505,1684)\) → \((-1,i,e\left(\frac{13}{72}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(7\)\(8\)\(10\)\(11\)\(13\)\(14\)\(16\)
\( \chi_{ 3723 }(1067, a) \) \(1\)\(1\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{67}{72}\right)\)\(e\left(\frac{17}{24}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{3}{8}\right)\)\(e\left(\frac{13}{72}\right)\)\(e\left(\frac{47}{72}\right)\)\(e\left(\frac{11}{72}\right)\)\(e\left(\frac{7}{9}\right)\)
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)
 
\( \chi_{ 3723 }(1067,a) \;\) at \(\;a = \) e.g. 2