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Properties

Label 10001.1614
Modulus $10001$
Conductor $10001$
Order $204$
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(10001, base_ring=CyclotomicField(204)) M = H._module chi = DirichletCharacter(H, M([68,27]))
 
Copy content gp:[g,chi] = znchar(Mod(1614, 10001))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("10001.1614");
 

Basic properties

Modulus: \(10001\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(10001\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(204\)
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 10001.dr

\(\chi_{10001}(8,\cdot)\) \(\chi_{10001}(154,\cdot)\) \(\chi_{10001}(283,\cdot)\) \(\chi_{10001}(592,\cdot)\) \(\chi_{10001}(721,\cdot)\) \(\chi_{10001}(794,\cdot)\) \(\chi_{10001}(811,\cdot)\) \(\chi_{10001}(940,\cdot)\) \(\chi_{10001}(957,\cdot)\) \(\chi_{10001}(1103,\cdot)\) \(\chi_{10001}(1378,\cdot)\) \(\chi_{10001}(1395,\cdot)\) \(\chi_{10001}(1468,\cdot)\) \(\chi_{10001}(1524,\cdot)\) \(\chi_{10001}(1614,\cdot)\) \(\chi_{10001}(1962,\cdot)\) \(\chi_{10001}(1979,\cdot)\) \(\chi_{10001}(2181,\cdot)\) \(\chi_{10001}(2327,\cdot)\) \(\chi_{10001}(2473,\cdot)\) \(\chi_{10001}(2765,\cdot)\) \(\chi_{10001}(2838,\cdot)\) \(\chi_{10001}(2984,\cdot)\) \(\chi_{10001}(3220,\cdot)\) \(\chi_{10001}(3349,\cdot)\) \(\chi_{10001}(3731,\cdot)\) \(\chi_{10001}(3804,\cdot)\) \(\chi_{10001}(4315,\cdot)\) \(\chi_{10001}(4590,\cdot)\) \(\chi_{10001}(5101,\cdot)\) ...

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: $\Q(\zeta_{204})$
Fixed field: Number field defined by a degree 204 polynomial (not computed)

Values on generators

\((4385,7812)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{9}{68}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 10001 }(1614, a) \) \(1\)\(1\)\(e\left(\frac{101}{102}\right)\)\(e\left(\frac{9}{68}\right)\)\(e\left(\frac{50}{51}\right)\)\(e\left(\frac{53}{204}\right)\)\(e\left(\frac{25}{204}\right)\)\(e\left(\frac{19}{34}\right)\)\(e\left(\frac{33}{34}\right)\)\(e\left(\frac{9}{34}\right)\)\(i\)\(e\left(\frac{49}{102}\right)\)
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)
 
\( \chi_{ 10001 }(1614,a) \;\) at \(\;a = \) e.g. 2