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Properties

Label 1441.444
Modulus $1441$
Conductor $1441$
Order $130$
Real no
Primitive yes
Minimal yes
Parity odd

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(1441, base_ring=CyclotomicField(130)) M = H._module chi = DirichletCharacter(H, M([26,115]))
 
Copy content gp:[g,chi] = znchar(Mod(444, 1441))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1441.444");
 

Basic properties

Modulus: \(1441\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1441\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(130\)
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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 1441.bt

\(\chi_{1441}(47,\cdot)\) \(\chi_{1441}(69,\cdot)\) \(\chi_{1441}(71,\cdot)\) \(\chi_{1441}(86,\cdot)\) \(\chi_{1441}(92,\cdot)\) \(\chi_{1441}(163,\cdot)\) \(\chi_{1441}(202,\cdot)\) \(\chi_{1441}(223,\cdot)\) \(\chi_{1441}(280,\cdot)\) \(\chi_{1441}(313,\cdot)\) \(\chi_{1441}(333,\cdot)\) \(\chi_{1441}(411,\cdot)\) \(\chi_{1441}(412,\cdot)\) \(\chi_{1441}(444,\cdot)\) \(\chi_{1441}(542,\cdot)\) \(\chi_{1441}(543,\cdot)\) \(\chi_{1441}(548,\cdot)\) \(\chi_{1441}(575,\cdot)\) \(\chi_{1441}(592,\cdot)\) \(\chi_{1441}(603,\cdot)\) \(\chi_{1441}(610,\cdot)\) \(\chi_{1441}(674,\cdot)\) \(\chi_{1441}(687,\cdot)\) \(\chi_{1441}(702,\cdot)\) \(\chi_{1441}(724,\cdot)\) \(\chi_{1441}(741,\cdot)\) \(\chi_{1441}(818,\cdot)\) \(\chi_{1441}(872,\cdot)\) \(\chi_{1441}(878,\cdot)\) \(\chi_{1441}(949,\cdot)\) ...

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: $\Q(\zeta_{65})$
Fixed field: Number field defined by a degree 130 polynomial (not computed)

Values on generators

\((1311,133)\) → \((e\left(\frac{1}{5}\right),e\left(\frac{23}{26}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 1441 }(444, a) \) \(-1\)\(1\)\(e\left(\frac{11}{130}\right)\)\(e\left(\frac{19}{65}\right)\)\(e\left(\frac{11}{65}\right)\)\(e\left(\frac{32}{65}\right)\)\(e\left(\frac{49}{130}\right)\)\(e\left(\frac{21}{65}\right)\)\(e\left(\frac{33}{130}\right)\)\(e\left(\frac{38}{65}\right)\)\(e\left(\frac{15}{26}\right)\)\(e\left(\frac{6}{13}\right)\)
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)
 
\( \chi_{ 1441 }(444,a) \;\) at \(\;a = \) e.g. 2