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Properties

Label 16359.2867
Modulus $16359$
Conductor $16359$
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(16359, base_ring=CyclotomicField(72)) M = H._module chi = DirichletCharacter(H, M([36,48,40,63]))
 
Copy content gp:[g,chi] = znchar(Mod(2867, 16359))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("16359.2867");
 

Basic properties

Modulus: \(16359\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(16359\)
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 16359.oa

\(\chi_{16359}(44,\cdot)\) \(\chi_{16359}(1145,\cdot)\) \(\chi_{16359}(1544,\cdot)\) \(\chi_{16359}(1859,\cdot)\) \(\chi_{16359}(2258,\cdot)\) \(\chi_{16359}(2867,\cdot)\) \(\chi_{16359}(3236,\cdot)\) \(\chi_{16359}(3266,\cdot)\) \(\chi_{16359}(3635,\cdot)\) \(\chi_{16359}(4811,\cdot)\) \(\chi_{16359}(5210,\cdot)\) \(\chi_{16359}(5450,\cdot)\) \(\chi_{16359}(5849,\cdot)\) \(\chi_{16359}(8255,\cdot)\) \(\chi_{16359}(8402,\cdot)\) \(\chi_{16359}(8654,\cdot)\) \(\chi_{16359}(8801,\cdot)\) \(\chi_{16359}(11846,\cdot)\) \(\chi_{16359}(12245,\cdot)\) \(\chi_{16359}(13913,\cdot)\) \(\chi_{16359}(14312,\cdot)\) \(\chi_{16359}(15635,\cdot)\) \(\chi_{16359}(16004,\cdot)\) \(\chi_{16359}(16034,\cdot)\)

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

\((10907,11686,13777,2794)\) → \((-1,e\left(\frac{2}{3}\right),e\left(\frac{5}{9}\right),e\left(\frac{7}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(5\)\(8\)\(10\)\(11\)\(13\)\(16\)\(17\)\(20\)
\( \chi_{ 16359 }(2867, a) \) \(1\)\(1\)\(e\left(\frac{5}{36}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{35}{36}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{11}{24}\right)\)\(e\left(\frac{65}{72}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{43}{72}\right)\)\(i\)
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)
 
\( \chi_{ 16359 }(2867,a) \;\) at \(\;a = \) e.g. 2