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Properties

Label 1883.195
Modulus $1883$
Conductor $1883$
Order $268$
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(1883, base_ring=CyclotomicField(268)) M = H._module chi = DirichletCharacter(H, M([134,191]))
 
Copy content gp:[g,chi] = znchar(Mod(195, 1883))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1883.195");
 

Basic properties

Modulus: \(1883\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1883\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(268\)
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 1883.s

\(\chi_{1883}(27,\cdot)\) \(\chi_{1883}(48,\cdot)\) \(\chi_{1883}(69,\cdot)\) \(\chi_{1883}(76,\cdot)\) \(\chi_{1883}(83,\cdot)\) \(\chi_{1883}(90,\cdot)\) \(\chi_{1883}(104,\cdot)\) \(\chi_{1883}(111,\cdot)\) \(\chi_{1883}(132,\cdot)\) \(\chi_{1883}(139,\cdot)\) \(\chi_{1883}(146,\cdot)\) \(\chi_{1883}(153,\cdot)\) \(\chi_{1883}(160,\cdot)\) \(\chi_{1883}(167,\cdot)\) \(\chi_{1883}(174,\cdot)\) \(\chi_{1883}(181,\cdot)\) \(\chi_{1883}(195,\cdot)\) \(\chi_{1883}(209,\cdot)\) \(\chi_{1883}(223,\cdot)\) \(\chi_{1883}(230,\cdot)\) \(\chi_{1883}(237,\cdot)\) \(\chi_{1883}(251,\cdot)\) \(\chi_{1883}(272,\cdot)\) \(\chi_{1883}(279,\cdot)\) \(\chi_{1883}(286,\cdot)\) \(\chi_{1883}(300,\cdot)\) \(\chi_{1883}(328,\cdot)\) \(\chi_{1883}(363,\cdot)\) \(\chi_{1883}(370,\cdot)\) \(\chi_{1883}(377,\cdot)\) ...

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: $\Q(\zeta_{268})$
Fixed field: Number field defined by a degree 268 polynomial (not computed)

Values on generators

\((808,540)\) → \((-1,e\left(\frac{191}{268}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(11\)\(12\)
\( \chi_{ 1883 }(195, a) \) \(1\)\(1\)\(e\left(\frac{191}{268}\right)\)\(e\left(\frac{49}{268}\right)\)\(e\left(\frac{57}{134}\right)\)\(e\left(\frac{99}{134}\right)\)\(e\left(\frac{60}{67}\right)\)\(e\left(\frac{37}{268}\right)\)\(e\left(\frac{49}{134}\right)\)\(e\left(\frac{121}{268}\right)\)\(e\left(\frac{123}{134}\right)\)\(e\left(\frac{163}{268}\right)\)
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)
 
\( \chi_{ 1883 }(195,a) \;\) at \(\;a = \) e.g. 2