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Properties

Label 4123.1640
Modulus $4123$
Conductor $4123$
Order $45$
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(4123, base_ring=CyclotomicField(90)) M = H._module chi = DirichletCharacter(H, M([30,70,48]))
 
Copy content gp:[g,chi] = znchar(Mod(1640, 4123))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4123.1640");
 

Basic properties

Modulus: \(4123\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(4123\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(45\)
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 4123.lp

\(\chi_{4123}(422,\cdot)\) \(\chi_{4123}(541,\cdot)\) \(\chi_{4123}(555,\cdot)\) \(\chi_{4123}(795,\cdot)\) \(\chi_{4123}(975,\cdot)\) \(\chi_{4123}(1012,\cdot)\) \(\chi_{4123}(1073,\cdot)\) \(\chi_{4123}(1206,\cdot)\) \(\chi_{4123}(1404,\cdot)\) \(\chi_{4123}(1507,\cdot)\) \(\chi_{4123}(1537,\cdot)\) \(\chi_{4123}(1619,\cdot)\) \(\chi_{4123}(1640,\cdot)\) \(\chi_{4123}(2270,\cdot)\) \(\chi_{4123}(2335,\cdot)\) \(\chi_{4123}(2704,\cdot)\) \(\chi_{4123}(2923,\cdot)\) \(\chi_{4123}(3056,\cdot)\) \(\chi_{4123}(3140,\cdot)\) \(\chi_{4123}(3273,\cdot)\) \(\chi_{4123}(3399,\cdot)\) \(\chi_{4123}(3854,\cdot)\) \(\chi_{4123}(4013,\cdot)\) \(\chi_{4123}(4071,\cdot)\)

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: $\Q(\zeta_{45})$
Fixed field: Number field defined by a degree 45 polynomial

Values on generators

\((1179,2605,1863)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{7}{9}\right),e\left(\frac{8}{15}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 4123 }(1640, a) \) \(1\)\(1\)\(e\left(\frac{11}{45}\right)\)\(e\left(\frac{44}{45}\right)\)\(e\left(\frac{22}{45}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{11}{15}\right)\)\(e\left(\frac{43}{45}\right)\)\(e\left(\frac{1}{45}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{7}{15}\right)\)
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)
 
\( \chi_{ 4123 }(1640,a) \;\) at \(\;a = \) e.g. 2