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Properties

Label 43907.89
Modulus $43907$
Conductor $43907$
Order $20746$
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(43907, base_ring=CyclotomicField(20746)) M = H._module chi = DirichletCharacter(H, M([11931,18469]))
 
Copy content gp:[g,chi] = znchar(Mod(89, 43907))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("43907.89");
 

Basic properties

Modulus: \(43907\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(43907\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(20746\)
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 43907.be

\(\chi_{43907}(5,\cdot)\) \(\chi_{43907}(14,\cdot)\) \(\chi_{43907}(15,\cdot)\) \(\chi_{43907}(19,\cdot)\) \(\chi_{43907}(20,\cdot)\) \(\chi_{43907}(34,\cdot)\) \(\chi_{43907}(43,\cdot)\) \(\chi_{43907}(53,\cdot)\) \(\chi_{43907}(56,\cdot)\) \(\chi_{43907}(57,\cdot)\) \(\chi_{43907}(60,\cdot)\) \(\chi_{43907}(66,\cdot)\) \(\chi_{43907}(67,\cdot)\) \(\chi_{43907}(74,\cdot)\) \(\chi_{43907}(76,\cdot)\) \(\chi_{43907}(79,\cdot)\) \(\chi_{43907}(80,\cdot)\) \(\chi_{43907}(88,\cdot)\) \(\chi_{43907}(89,\cdot)\) \(\chi_{43907}(97,\cdot)\) \(\chi_{43907}(102,\cdot)\) \(\chi_{43907}(103,\cdot)\) \(\chi_{43907}(107,\cdot)\) \(\chi_{43907}(122,\cdot)\) \(\chi_{43907}(125,\cdot)\) \(\chi_{43907}(126,\cdot)\) \(\chi_{43907}(129,\cdot)\) \(\chi_{43907}(135,\cdot)\) \(\chi_{43907}(136,\cdot)\) \(\chi_{43907}(143,\cdot)\) ...

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: $\Q(\zeta_{10373})$
Fixed field: Number field defined by a degree 20746 polynomial (not computed)

Values on generators

\((16933,39676)\) → \((e\left(\frac{291}{506}\right),e\left(\frac{73}{82}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 43907 }(89, a) \) \(1\)\(1\)\(e\left(\frac{18879}{20746}\right)\)\(e\left(\frac{3103}{10373}\right)\)\(e\left(\frac{8506}{10373}\right)\)\(e\left(\frac{6345}{10373}\right)\)\(e\left(\frac{4339}{20746}\right)\)\(e\left(\frac{6425}{20746}\right)\)\(e\left(\frac{15145}{20746}\right)\)\(e\left(\frac{6206}{10373}\right)\)\(e\left(\frac{10823}{20746}\right)\)\(e\left(\frac{12141}{20746}\right)\)
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)
 
\( \chi_{ 43907 }(89,a) \;\) at \(\;a = \) e.g. 2