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Properties

Label 1279.178
Modulus $1279$
Conductor $1279$
Order $426$
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(1279, base_ring=CyclotomicField(426)) M = H._module chi = DirichletCharacter(H, M([421]))
 
Copy content gp:[g,chi] = znchar(Mod(178, 1279))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1279.178");
 

Basic properties

Modulus: \(1279\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1279\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(426\)
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 1279.j

\(\chi_{1279}(11,\cdot)\) \(\chi_{1279}(12,\cdot)\) \(\chi_{1279}(19,\cdot)\) \(\chi_{1279}(27,\cdot)\) \(\chi_{1279}(30,\cdot)\) \(\chi_{1279}(42,\cdot)\) \(\chi_{1279}(58,\cdot)\) \(\chi_{1279}(71,\cdot)\) \(\chi_{1279}(75,\cdot)\) \(\chi_{1279}(88,\cdot)\) \(\chi_{1279}(96,\cdot)\) \(\chi_{1279}(105,\cdot)\) \(\chi_{1279}(111,\cdot)\) \(\chi_{1279}(124,\cdot)\) \(\chi_{1279}(134,\cdot)\) \(\chi_{1279}(137,\cdot)\) \(\chi_{1279}(145,\cdot)\) \(\chi_{1279}(147,\cdot)\) \(\chi_{1279}(149,\cdot)\) \(\chi_{1279}(177,\cdot)\) \(\chi_{1279}(178,\cdot)\) \(\chi_{1279}(198,\cdot)\) \(\chi_{1279}(202,\cdot)\) \(\chi_{1279}(203,\cdot)\) \(\chi_{1279}(216,\cdot)\) \(\chi_{1279}(220,\cdot)\) \(\chi_{1279}(221,\cdot)\) \(\chi_{1279}(240,\cdot)\) \(\chi_{1279}(262,\cdot)\) \(\chi_{1279}(279,\cdot)\) ...

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: $\Q(\zeta_{213})$
Fixed field: Number field defined by a degree 426 polynomial (not computed)

Values on generators

\(3\) → \(e\left(\frac{421}{426}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1279 }(178, a) \) \(-1\)\(1\)\(e\left(\frac{98}{213}\right)\)\(e\left(\frac{421}{426}\right)\)\(e\left(\frac{196}{213}\right)\)\(e\left(\frac{68}{213}\right)\)\(e\left(\frac{191}{426}\right)\)\(e\left(\frac{137}{213}\right)\)\(e\left(\frac{27}{71}\right)\)\(e\left(\frac{208}{213}\right)\)\(e\left(\frac{166}{213}\right)\)\(e\left(\frac{47}{142}\right)\)
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)
 
\( \chi_{ 1279 }(178,a) \;\) at \(\;a = \) e.g. 2