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Properties

Label 1429.76
Modulus $1429$
Conductor $1429$
Order $119$
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(1429, base_ring=CyclotomicField(238)) M = H._module chi = DirichletCharacter(H, M([4]))
 
Copy content gp:[g,chi] = znchar(Mod(76, 1429))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1429.76");
 

Basic properties

Modulus: \(1429\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1429\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(119\)
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 1429.r

\(\chi_{1429}(5,\cdot)\) \(\chi_{1429}(12,\cdot)\) \(\chi_{1429}(25,\cdot)\) \(\chi_{1429}(51,\cdot)\) \(\chi_{1429}(52,\cdot)\) \(\chi_{1429}(60,\cdot)\) \(\chi_{1429}(63,\cdot)\) \(\chi_{1429}(66,\cdot)\) \(\chi_{1429}(69,\cdot)\) \(\chi_{1429}(71,\cdot)\) \(\chi_{1429}(76,\cdot)\) \(\chi_{1429}(125,\cdot)\) \(\chi_{1429}(144,\cdot)\) \(\chi_{1429}(146,\cdot)\) \(\chi_{1429}(174,\cdot)\) \(\chi_{1429}(186,\cdot)\) \(\chi_{1429}(199,\cdot)\) \(\chi_{1429}(221,\cdot)\) \(\chi_{1429}(255,\cdot)\) \(\chi_{1429}(260,\cdot)\) \(\chi_{1429}(262,\cdot)\) \(\chi_{1429}(267,\cdot)\) \(\chi_{1429}(273,\cdot)\) \(\chi_{1429}(286,\cdot)\) \(\chi_{1429}(296,\cdot)\) \(\chi_{1429}(299,\cdot)\) \(\chi_{1429}(300,\cdot)\) \(\chi_{1429}(315,\cdot)\) \(\chi_{1429}(323,\cdot)\) \(\chi_{1429}(330,\cdot)\) ...

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: $\Q(\zeta_{119})$
Fixed field: Number field defined by a degree 119 polynomial (not computed)

Values on generators

\(6\) → \(e\left(\frac{2}{119}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1429 }(76, a) \) \(1\)\(1\)\(e\left(\frac{6}{7}\right)\)\(e\left(\frac{19}{119}\right)\)\(e\left(\frac{5}{7}\right)\)\(e\left(\frac{111}{119}\right)\)\(e\left(\frac{2}{119}\right)\)\(e\left(\frac{45}{119}\right)\)\(e\left(\frac{4}{7}\right)\)\(e\left(\frac{38}{119}\right)\)\(e\left(\frac{94}{119}\right)\)\(e\left(\frac{64}{119}\right)\)
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)
 
\( \chi_{ 1429 }(76,a) \;\) at \(\;a = \) e.g. 2