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Properties

Label 5000.531
Modulus $5000$
Conductor $5000$
Order $250$
Real no
Primitive yes
Minimal yes
Parity odd

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(5000, base_ring=CyclotomicField(250)) M = H._module chi = DirichletCharacter(H, M([125,125,174]))
 
Copy content gp:[g,chi] = znchar(Mod(531, 5000))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("5000.531");
 

Basic properties

Modulus: \(5000\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(5000\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(250\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 5000.bl

\(\chi_{5000}(11,\cdot)\) \(\chi_{5000}(91,\cdot)\) \(\chi_{5000}(131,\cdot)\) \(\chi_{5000}(171,\cdot)\) \(\chi_{5000}(211,\cdot)\) \(\chi_{5000}(291,\cdot)\) \(\chi_{5000}(331,\cdot)\) \(\chi_{5000}(371,\cdot)\) \(\chi_{5000}(411,\cdot)\) \(\chi_{5000}(491,\cdot)\) \(\chi_{5000}(531,\cdot)\) \(\chi_{5000}(571,\cdot)\) \(\chi_{5000}(611,\cdot)\) \(\chi_{5000}(691,\cdot)\) \(\chi_{5000}(731,\cdot)\) \(\chi_{5000}(771,\cdot)\) \(\chi_{5000}(811,\cdot)\) \(\chi_{5000}(891,\cdot)\) \(\chi_{5000}(931,\cdot)\) \(\chi_{5000}(971,\cdot)\) \(\chi_{5000}(1011,\cdot)\) \(\chi_{5000}(1091,\cdot)\) \(\chi_{5000}(1131,\cdot)\) \(\chi_{5000}(1171,\cdot)\) \(\chi_{5000}(1211,\cdot)\) \(\chi_{5000}(1291,\cdot)\) \(\chi_{5000}(1331,\cdot)\) \(\chi_{5000}(1371,\cdot)\) \(\chi_{5000}(1411,\cdot)\) \(\chi_{5000}(1491,\cdot)\) ...

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: $\Q(\zeta_{125})$
Fixed field: Number field defined by a degree 250 polynomial (not computed)

Values on generators

\((3751,2501,4377)\) → \((-1,-1,e\left(\frac{87}{125}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 5000 }(531, a) \) \(-1\)\(1\)\(e\left(\frac{59}{125}\right)\)\(e\left(\frac{3}{50}\right)\)\(e\left(\frac{118}{125}\right)\)\(e\left(\frac{37}{125}\right)\)\(e\left(\frac{61}{250}\right)\)\(e\left(\frac{51}{125}\right)\)\(e\left(\frac{116}{125}\right)\)\(e\left(\frac{133}{250}\right)\)\(e\left(\frac{119}{250}\right)\)\(e\left(\frac{52}{125}\right)\)
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)
 
\( \chi_{ 5000 }(531,a) \;\) at \(\;a = \) e.g. 2