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Properties

Label 5125.622
Modulus $5125$
Conductor $5125$
Order $200$
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(5125, base_ring=CyclotomicField(200)) M = H._module chi = DirichletCharacter(H, M([114,195]))
 
Copy content gp:[g,chi] = znchar(Mod(622, 5125))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5125.622");
 

Basic properties

Modulus: \(5125\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(5125\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(200\)
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 5125.fx

\(\chi_{5125}(112,\cdot)\) \(\chi_{5125}(152,\cdot)\) \(\chi_{5125}(158,\cdot)\) \(\chi_{5125}(188,\cdot)\) \(\chi_{5125}(192,\cdot)\) \(\chi_{5125}(263,\cdot)\) \(\chi_{5125}(462,\cdot)\) \(\chi_{5125}(548,\cdot)\) \(\chi_{5125}(622,\cdot)\) \(\chi_{5125}(627,\cdot)\) \(\chi_{5125}(678,\cdot)\) \(\chi_{5125}(723,\cdot)\) \(\chi_{5125}(772,\cdot)\) \(\chi_{5125}(792,\cdot)\) \(\chi_{5125}(908,\cdot)\) \(\chi_{5125}(1003,\cdot)\) \(\chi_{5125}(1137,\cdot)\) \(\chi_{5125}(1177,\cdot)\) \(\chi_{5125}(1183,\cdot)\) \(\chi_{5125}(1213,\cdot)\) \(\chi_{5125}(1217,\cdot)\) \(\chi_{5125}(1288,\cdot)\) \(\chi_{5125}(1487,\cdot)\) \(\chi_{5125}(1573,\cdot)\) \(\chi_{5125}(1647,\cdot)\) \(\chi_{5125}(1652,\cdot)\) \(\chi_{5125}(1703,\cdot)\) \(\chi_{5125}(1748,\cdot)\) \(\chi_{5125}(1797,\cdot)\) \(\chi_{5125}(1817,\cdot)\) ...

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: $\Q(\zeta_{200})$
Fixed field: Number field defined by a degree 200 polynomial (not computed)

Values on generators

\((2502,2876)\) → \((e\left(\frac{57}{100}\right),e\left(\frac{39}{40}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 5125 }(622, a) \) \(1\)\(1\)\(e\left(\frac{23}{25}\right)\)\(e\left(\frac{123}{200}\right)\)\(e\left(\frac{21}{25}\right)\)\(e\left(\frac{107}{200}\right)\)\(e\left(\frac{19}{40}\right)\)\(e\left(\frac{19}{25}\right)\)\(e\left(\frac{23}{100}\right)\)\(e\left(\frac{49}{200}\right)\)\(e\left(\frac{91}{200}\right)\)\(e\left(\frac{91}{200}\right)\)
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)
 
\( \chi_{ 5125 }(622,a) \;\) at \(\;a = \) e.g. 2