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Properties

Label 62656.1821
Modulus $62656$
Conductor $62656$
Order $880$
Real no
Primitive yes
Minimal yes
Parity even

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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(62656, base_ring=CyclotomicField(880)) M = H._module chi = DirichletCharacter(H, M([0,605,792,210]))
 
Copy content gp:[g,chi] = znchar(Mod(1821, 62656))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("62656.1821");
 

Basic properties

Modulus: \(62656\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(62656\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(880\)
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 62656.sc

\(\chi_{62656}(61,\cdot)\) \(\chi_{62656}(293,\cdot)\) \(\chi_{62656}(349,\cdot)\) \(\chi_{62656}(557,\cdot)\) \(\chi_{62656}(1757,\cdot)\) \(\chi_{62656}(1821,\cdot)\) \(\chi_{62656}(1965,\cdot)\) \(\chi_{62656}(2317,\cdot)\) \(\chi_{62656}(2525,\cdot)\) \(\chi_{62656}(2637,\cdot)\) \(\chi_{62656}(2789,\cdot)\) \(\chi_{62656}(2845,\cdot)\) \(\chi_{62656}(3077,\cdot)\) \(\chi_{62656}(3141,\cdot)\) \(\chi_{62656}(3197,\cdot)\) \(\chi_{62656}(3341,\cdot)\) \(\chi_{62656}(3405,\cdot)\) \(\chi_{62656}(3445,\cdot)\) \(\chi_{62656}(3797,\cdot)\) \(\chi_{62656}(4485,\cdot)\) \(\chi_{62656}(4605,\cdot)\) \(\chi_{62656}(4813,\cdot)\) \(\chi_{62656}(4837,\cdot)\) \(\chi_{62656}(4941,\cdot)\) \(\chi_{62656}(5013,\cdot)\) \(\chi_{62656}(5101,\cdot)\) \(\chi_{62656}(5165,\cdot)\) \(\chi_{62656}(5453,\cdot)\) \(\chi_{62656}(5485,\cdot)\) \(\chi_{62656}(5693,\cdot)\) ...

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: $\Q(\zeta_{880})$
Fixed field: Number field defined by a degree 880 polynomial (not computed)

Values on generators

\((9791,43077,28481,15489)\) → \((1,e\left(\frac{11}{16}\right),e\left(\frac{9}{10}\right),e\left(\frac{21}{88}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(13\)\(15\)\(17\)\(19\)\(21\)\(23\)
\( \chi_{ 62656 }(1821, a) \) \(1\)\(1\)\(e\left(\frac{441}{880}\right)\)\(e\left(\frac{873}{880}\right)\)\(e\left(\frac{111}{220}\right)\)\(e\left(\frac{1}{440}\right)\)\(e\left(\frac{617}{880}\right)\)\(e\left(\frac{217}{440}\right)\)\(e\left(\frac{43}{55}\right)\)\(e\left(\frac{761}{880}\right)\)\(e\left(\frac{1}{176}\right)\)\(e\left(\frac{5}{22}\right)\)
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)
 
\( \chi_{ 62656 }(1821,a) \;\) at \(\;a = \) e.g. 2