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Properties

Label 1445.109
Modulus $1445$
Conductor $1445$
Order $272$
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(1445, base_ring=CyclotomicField(272)) M = H._module chi = DirichletCharacter(H, M([136,203]))
 
Copy content gp:[g,chi] = znchar(Mod(109, 1445))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1445.109");
 

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: odd
Copy content comment:Parity
 
Copy content sage:chi.is_odd()
 
Copy content gp:zncharisodd(g,chi)
 
Copy content magma:IsOdd(chi);
 

Galois orbit 1445.bi

\(\chi_{1445}(14,\cdot)\) \(\chi_{1445}(24,\cdot)\) \(\chi_{1445}(29,\cdot)\) \(\chi_{1445}(39,\cdot)\) \(\chi_{1445}(44,\cdot)\) \(\chi_{1445}(54,\cdot)\) \(\chi_{1445}(74,\cdot)\) \(\chi_{1445}(79,\cdot)\) \(\chi_{1445}(99,\cdot)\) \(\chi_{1445}(109,\cdot)\) \(\chi_{1445}(114,\cdot)\) \(\chi_{1445}(124,\cdot)\) \(\chi_{1445}(129,\cdot)\) \(\chi_{1445}(139,\cdot)\) \(\chi_{1445}(159,\cdot)\) \(\chi_{1445}(164,\cdot)\) \(\chi_{1445}(184,\cdot)\) \(\chi_{1445}(194,\cdot)\) \(\chi_{1445}(199,\cdot)\) \(\chi_{1445}(209,\cdot)\) \(\chi_{1445}(244,\cdot)\) \(\chi_{1445}(269,\cdot)\) \(\chi_{1445}(279,\cdot)\) \(\chi_{1445}(284,\cdot)\) \(\chi_{1445}(294,\cdot)\) \(\chi_{1445}(299,\cdot)\) \(\chi_{1445}(309,\cdot)\) \(\chi_{1445}(334,\cdot)\) \(\chi_{1445}(369,\cdot)\) \(\chi_{1445}(379,\cdot)\) ...

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

\((1157,581)\) → \((-1,e\left(\frac{203}{272}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(7\)\(8\)\(9\)\(11\)\(12\)\(13\)
\( \chi_{ 1445 }(109, a) \) \(-1\)\(1\)\(e\left(\frac{41}{136}\right)\)\(e\left(\frac{67}{272}\right)\)\(e\left(\frac{41}{68}\right)\)\(e\left(\frac{149}{272}\right)\)\(e\left(\frac{49}{272}\right)\)\(e\left(\frac{123}{136}\right)\)\(e\left(\frac{67}{136}\right)\)\(e\left(\frac{45}{272}\right)\)\(e\left(\frac{231}{272}\right)\)\(e\left(\frac{53}{68}\right)\)
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)
 
\( \chi_{ 1445 }(109,a) \;\) at \(\;a = \) e.g. 2