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Properties

Label 1445.12
Modulus $1445$
Conductor $1445$
Order $272$
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(1445, base_ring=CyclotomicField(272)) M = H._module chi = DirichletCharacter(H, M([68,109]))
 
Copy content gp:[g,chi] = znchar(Mod(12, 1445))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1445.12");
 

Basic properties

Modulus: \(1445\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1445\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(272\)
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 1445.bg

\(\chi_{1445}(12,\cdot)\) \(\chi_{1445}(22,\cdot)\) \(\chi_{1445}(23,\cdot)\) \(\chi_{1445}(28,\cdot)\) \(\chi_{1445}(37,\cdot)\) \(\chi_{1445}(58,\cdot)\) \(\chi_{1445}(78,\cdot)\) \(\chi_{1445}(82,\cdot)\) \(\chi_{1445}(97,\cdot)\) \(\chi_{1445}(107,\cdot)\) \(\chi_{1445}(108,\cdot)\) \(\chi_{1445}(113,\cdot)\) \(\chi_{1445}(122,\cdot)\) \(\chi_{1445}(143,\cdot)\) \(\chi_{1445}(163,\cdot)\) \(\chi_{1445}(167,\cdot)\) \(\chi_{1445}(182,\cdot)\) \(\chi_{1445}(192,\cdot)\) \(\chi_{1445}(193,\cdot)\) \(\chi_{1445}(198,\cdot)\) \(\chi_{1445}(207,\cdot)\) \(\chi_{1445}(228,\cdot)\) \(\chi_{1445}(248,\cdot)\) \(\chi_{1445}(252,\cdot)\) \(\chi_{1445}(267,\cdot)\) \(\chi_{1445}(277,\cdot)\) \(\chi_{1445}(278,\cdot)\) \(\chi_{1445}(283,\cdot)\) \(\chi_{1445}(292,\cdot)\) \(\chi_{1445}(313,\cdot)\) ...

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: $\Q(\zeta_{272})$
Fixed field: Number field defined by a degree 272 polynomial (not computed)

Values on generators

\((1157,581)\) → \((i,e\left(\frac{109}{272}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 1445 }(12, a) \) \(1\)\(1\)\(e\left(\frac{53}{136}\right)\)\(e\left(\frac{41}{272}\right)\)\(e\left(\frac{53}{68}\right)\)\(e\left(\frac{147}{272}\right)\)\(e\left(\frac{99}{272}\right)\)\(e\left(\frac{23}{136}\right)\)\(e\left(\frac{41}{136}\right)\)\(e\left(\frac{59}{272}\right)\)\(e\left(\frac{253}{272}\right)\)\(e\left(\frac{5}{17}\right)\)
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)
 
\( \chi_{ 1445 }(12,a) \;\) at \(\;a = \) e.g. 2