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Properties

Label 72013.508
Modulus $72013$
Conductor $72013$
Order $3300$
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(72013, base_ring=CyclotomicField(3300)) M = H._module chi = DirichletCharacter(H, M([300,2090,2277]))
 
Copy content gp:[g,chi] = znchar(Mod(508, 72013))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("72013.508");
 

Basic properties

Modulus: \(72013\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(72013\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(3300\)
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 72013.vk

\(\chi_{72013}(141,\cdot)\) \(\chi_{72013}(210,\cdot)\) \(\chi_{72013}(265,\cdot)\) \(\chi_{72013}(353,\cdot)\) \(\chi_{72013}(358,\cdot)\) \(\chi_{72013}(416,\cdot)\) \(\chi_{72013}(446,\cdot)\) \(\chi_{72013}(455,\cdot)\) \(\chi_{72013}(508,\cdot)\) \(\chi_{72013}(725,\cdot)\) \(\chi_{72013}(854,\cdot)\) \(\chi_{72013}(947,\cdot)\) \(\chi_{72013}(951,\cdot)\) \(\chi_{72013}(1071,\cdot)\) \(\chi_{72013}(1315,\cdot)\) \(\chi_{72013}(1406,\cdot)\) \(\chi_{72013}(1407,\cdot)\) \(\chi_{72013}(1530,\cdot)\)

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: $\Q(\zeta_{3300})$
Fixed field: Number field defined by a degree 3300 polynomial (not computed)

Values on generators

\((28180,69691,12122)\) → \((e\left(\frac{1}{11}\right),e\left(\frac{19}{30}\right),e\left(\frac{69}{100}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 72013 }(508, a) \) \(1\)\(1\)\(e\left(\frac{79}{1100}\right)\)\(e\left(\frac{2303}{3300}\right)\)\(e\left(\frac{79}{550}\right)\)\(e\left(\frac{262}{825}\right)\)\(e\left(\frac{127}{165}\right)\)\(e\left(\frac{2213}{3300}\right)\)\(e\left(\frac{237}{1100}\right)\)\(e\left(\frac{653}{1650}\right)\)\(e\left(\frac{257}{660}\right)\)\(e\left(\frac{1171}{3300}\right)\)
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)
 
\( \chi_{ 72013 }(508,a) \;\) at \(\;a = \) e.g. 2