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Properties

Label 6500.4221
Modulus $6500$
Conductor $1625$
Order $75$
Real no
Primitive no
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(6500, base_ring=CyclotomicField(150)) M = H._module chi = DirichletCharacter(H, M([0,18,100]))
 
Copy content gp:[g,chi] = znchar(Mod(4221, 6500))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("6500.4221");
 

Basic properties

Modulus: \(6500\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1625\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(75\)
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: no, induced from \(\chi_{1625}(971,\cdot)\)
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 6500.dk

\(\chi_{6500}(61,\cdot)\) \(\chi_{6500}(81,\cdot)\) \(\chi_{6500}(321,\cdot)\) \(\chi_{6500}(341,\cdot)\) \(\chi_{6500}(581,\cdot)\) \(\chi_{6500}(841,\cdot)\) \(\chi_{6500}(861,\cdot)\) \(\chi_{6500}(1121,\cdot)\) \(\chi_{6500}(1361,\cdot)\) \(\chi_{6500}(1381,\cdot)\) \(\chi_{6500}(1621,\cdot)\) \(\chi_{6500}(1641,\cdot)\) \(\chi_{6500}(1881,\cdot)\) \(\chi_{6500}(2141,\cdot)\) \(\chi_{6500}(2161,\cdot)\) \(\chi_{6500}(2421,\cdot)\) \(\chi_{6500}(2661,\cdot)\) \(\chi_{6500}(2681,\cdot)\) \(\chi_{6500}(2921,\cdot)\) \(\chi_{6500}(2941,\cdot)\) \(\chi_{6500}(3181,\cdot)\) \(\chi_{6500}(3441,\cdot)\) \(\chi_{6500}(3461,\cdot)\) \(\chi_{6500}(3721,\cdot)\) \(\chi_{6500}(3961,\cdot)\) \(\chi_{6500}(3981,\cdot)\) \(\chi_{6500}(4221,\cdot)\) \(\chi_{6500}(4241,\cdot)\) \(\chi_{6500}(4481,\cdot)\) \(\chi_{6500}(4741,\cdot)\) ...

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: $\Q(\zeta_{75})$
Copy content comment:Field of values of chi
 
Copy content sage:CyclotomicField(chi.multiplicative_order())
 
Copy content gp:nfinit(polcyclo(charorder(g,chi)))
 
Copy content magma:CyclotomicField(Order(chi));
 
Fixed field: Number field defined by a degree 75 polynomial
Copy content comment:Fixed field
 
Copy content sage:chi.fixed_field()
 

Values on generators

\((3251,5877,5501)\) → \((1,e\left(\frac{3}{25}\right),e\left(\frac{2}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(17\)\(19\)\(21\)\(23\)\(27\)\(29\)
\( \chi_{ 6500 }(4221, a) \) \(1\)\(1\)\(e\left(\frac{38}{75}\right)\)\(e\left(\frac{8}{15}\right)\)\(e\left(\frac{1}{75}\right)\)\(e\left(\frac{59}{75}\right)\)\(e\left(\frac{7}{75}\right)\)\(e\left(\frac{37}{75}\right)\)\(e\left(\frac{1}{25}\right)\)\(e\left(\frac{29}{75}\right)\)\(e\left(\frac{13}{25}\right)\)\(e\left(\frac{8}{75}\right)\)
Copy content comment:Value of chi at x
 
Copy content sage:chi(x) # x integer
 
Copy content gp:chareval(g,chi,x) \\ x integer, value in Q/Z
 
Copy content magma:chi(x)
 
\( \chi_{ 6500 }(4221,a) \;\) at \(\;a = \) e.g. 2