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Properties

Label 212800.13123
Modulus $212800$
Conductor $212800$
Order $720$
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(212800, base_ring=CyclotomicField(720)) M = H._module chi = DirichletCharacter(H, M([360,135,396,600,200]))
 
Copy content gp:[g,chi] = znchar(Mod(13123, 212800))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("212800.13123");
 

Basic properties

Modulus: \(212800\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(212800\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(720\)
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 212800.dhm

\(\chi_{212800}(3,\cdot)\) \(\chi_{212800}(1067,\cdot)\) \(\chi_{212800}(1123,\cdot)\) \(\chi_{212800}(2187,\cdot)\) \(\chi_{212800}(2483,\cdot)\) \(\chi_{212800}(3547,\cdot)\) \(\chi_{212800}(8963,\cdot)\) \(\chi_{212800}(10027,\cdot)\) \(\chi_{212800}(10883,\cdot)\) \(\chi_{212800}(11763,\cdot)\) \(\chi_{212800}(11947,\cdot)\) \(\chi_{212800}(12827,\cdot)\) \(\chi_{212800}(13123,\cdot)\) \(\chi_{212800}(14187,\cdot)\) \(\chi_{212800}(16483,\cdot)\) \(\chi_{212800}(17547,\cdot)\) \(\chi_{212800}(19603,\cdot)\) \(\chi_{212800}(20667,\cdot)\) \(\chi_{212800}(21283,\cdot)\) \(\chi_{212800}(21523,\cdot)\) \(\chi_{212800}(22347,\cdot)\) \(\chi_{212800}(22403,\cdot)\) \(\chi_{212800}(22587,\cdot)\) \(\chi_{212800}(23467,\cdot)\) \(\chi_{212800}(23763,\cdot)\) \(\chi_{212800}(24827,\cdot)\) \(\chi_{212800}(27123,\cdot)\) \(\chi_{212800}(28187,\cdot)\) \(\chi_{212800}(31923,\cdot)\) \(\chi_{212800}(32163,\cdot)\) ...

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: $\Q(\zeta_{720})$
Fixed field: Number field defined by a degree 720 polynomial (not computed)

Values on generators

\((73151,66501,195777,152001,190401)\) → \((-1,e\left(\frac{3}{16}\right),e\left(\frac{11}{20}\right),e\left(\frac{5}{6}\right),e\left(\frac{5}{18}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(9\)\(11\)\(13\)\(17\)\(23\)\(27\)\(29\)\(31\)\(33\)
\( \chi_{ 212800 }(13123, a) \) \(1\)\(1\)\(e\left(\frac{257}{720}\right)\)\(e\left(\frac{257}{360}\right)\)\(e\left(\frac{217}{240}\right)\)\(e\left(\frac{109}{720}\right)\)\(e\left(\frac{1}{90}\right)\)\(e\left(\frac{143}{360}\right)\)\(e\left(\frac{17}{240}\right)\)\(e\left(\frac{637}{720}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{47}{180}\right)\)
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)
 
\( \chi_{ 212800 }(13123,a) \;\) at \(\;a = \) e.g. 2