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Properties

Label 220669.3965
Modulus $220669$
Conductor $220669$
Order $1480$
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(220669, base_ring=CyclotomicField(1480)) M = H._module chi = DirichletCharacter(H, M([470,113]))
 
Copy content gp:[g,chi] = znchar(Mod(3965, 220669))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("220669.3965");
 

Basic properties

Modulus: \(220669\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(220669\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(1480\)
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 220669.bog

\(\chi_{220669}(839,\cdot)\) \(\chi_{220669}(960,\cdot)\) \(\chi_{220669}(1009,\cdot)\) \(\chi_{220669}(2124,\cdot)\) \(\chi_{220669}(3039,\cdot)\) \(\chi_{220669}(3379,\cdot)\) \(\chi_{220669}(3525,\cdot)\) \(\chi_{220669}(3589,\cdot)\) \(\chi_{220669}(3871,\cdot)\) \(\chi_{220669}(3965,\cdot)\) \(\chi_{220669}(4462,\cdot)\) \(\chi_{220669}(4567,\cdot)\) \(\chi_{220669}(4778,\cdot)\) \(\chi_{220669}(5472,\cdot)\) \(\chi_{220669}(5630,\cdot)\) \(\chi_{220669}(5882,\cdot)\) \(\chi_{220669}(6268,\cdot)\) \(\chi_{220669}(6541,\cdot)\) \(\chi_{220669}(6645,\cdot)\) \(\chi_{220669}(7427,\cdot)\) \(\chi_{220669}(7559,\cdot)\) \(\chi_{220669}(7808,\cdot)\) \(\chi_{220669}(8216,\cdot)\) \(\chi_{220669}(8273,\cdot)\) \(\chi_{220669}(8544,\cdot)\) \(\chi_{220669}(8972,\cdot)\) \(\chi_{220669}(10470,\cdot)\) \(\chi_{220669}(11342,\cdot)\) \(\chi_{220669}(11452,\cdot)\) \(\chi_{220669}(12158,\cdot)\) ...

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: $\Q(\zeta_{1480})$
Fixed field: Number field defined by a degree 1480 polynomial (not computed)

Values on generators

\((48874,122926)\) → \((e\left(\frac{47}{148}\right),e\left(\frac{113}{1480}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 220669 }(3965, a) \) \(1\)\(1\)\(e\left(\frac{97}{740}\right)\)\(e\left(\frac{1043}{1480}\right)\)\(e\left(\frac{97}{370}\right)\)\(e\left(\frac{287}{740}\right)\)\(e\left(\frac{1237}{1480}\right)\)\(e\left(\frac{273}{370}\right)\)\(e\left(\frac{291}{740}\right)\)\(e\left(\frac{303}{740}\right)\)\(e\left(\frac{96}{185}\right)\)\(e\left(\frac{193}{296}\right)\)
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)
 
\( \chi_{ 220669 }(3965,a) \;\) at \(\;a = \) e.g. 2