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Properties

Label 1343.107
Modulus $1343$
Conductor $1343$
Order $624$
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(1343, base_ring=CyclotomicField(624)) M = H._module chi = DirichletCharacter(H, M([195,488]))
 
Copy content gp:[g,chi] = znchar(Mod(107, 1343))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1343.107");
 

Basic properties

Modulus: \(1343\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1343\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(624\)
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 1343.bm

\(\chi_{1343}(3,\cdot)\) \(\chi_{1343}(6,\cdot)\) \(\chi_{1343}(7,\cdot)\) \(\chi_{1343}(28,\cdot)\) \(\chi_{1343}(29,\cdot)\) \(\chi_{1343}(37,\cdot)\) \(\chi_{1343}(39,\cdot)\) \(\chi_{1343}(48,\cdot)\) \(\chi_{1343}(54,\cdot)\) \(\chi_{1343}(63,\cdot)\) \(\chi_{1343}(74,\cdot)\) \(\chi_{1343}(75,\cdot)\) \(\chi_{1343}(82,\cdot)\) \(\chi_{1343}(107,\cdot)\) \(\chi_{1343}(108,\cdot)\) \(\chi_{1343}(109,\cdot)\) \(\chi_{1343}(113,\cdot)\) \(\chi_{1343}(114,\cdot)\) \(\chi_{1343}(116,\cdot)\) \(\chi_{1343}(122,\cdot)\) \(\chi_{1343}(126,\cdot)\) \(\chi_{1343}(133,\cdot)\) \(\chi_{1343}(139,\cdot)\) \(\chi_{1343}(142,\cdot)\) \(\chi_{1343}(147,\cdot)\) \(\chi_{1343}(156,\cdot)\) \(\chi_{1343}(164,\cdot)\) \(\chi_{1343}(165,\cdot)\) \(\chi_{1343}(192,\cdot)\) \(\chi_{1343}(193,\cdot)\) ...

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: $\Q(\zeta_{624})$
Fixed field: Number field defined by a degree 624 polynomial (not computed)

Values on generators

\((870,477)\) → \((e\left(\frac{5}{16}\right),e\left(\frac{61}{78}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 1343 }(107, a) \) \(1\)\(1\)\(e\left(\frac{157}{312}\right)\)\(e\left(\frac{59}{624}\right)\)\(e\left(\frac{1}{156}\right)\)\(e\left(\frac{31}{624}\right)\)\(e\left(\frac{373}{624}\right)\)\(e\left(\frac{553}{624}\right)\)\(e\left(\frac{53}{104}\right)\)\(e\left(\frac{59}{312}\right)\)\(e\left(\frac{115}{208}\right)\)\(e\left(\frac{229}{624}\right)\)
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)
 
\( \chi_{ 1343 }(107,a) \;\) at \(\;a = \) e.g. 2