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Properties

Label 1309.39
Modulus $1309$
Conductor $1309$
Order $240$
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(1309, base_ring=CyclotomicField(240)) M = H._module chi = DirichletCharacter(H, M([160,216,75]))
 
Copy content gp:[g,chi] = znchar(Mod(39, 1309))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1309.39");
 

Basic properties

Modulus: \(1309\)
Copy content comment:Modulus
 
Copy content sage:chi.modulus()
 
Copy content gp:g[1][1]
 
Copy content magma:Modulus(chi);
 
Conductor: \(1309\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(240\)
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 1309.cy

\(\chi_{1309}(39,\cdot)\) \(\chi_{1309}(46,\cdot)\) \(\chi_{1309}(74,\cdot)\) \(\chi_{1309}(79,\cdot)\) \(\chi_{1309}(95,\cdot)\) \(\chi_{1309}(107,\cdot)\) \(\chi_{1309}(116,\cdot)\) \(\chi_{1309}(156,\cdot)\) \(\chi_{1309}(184,\cdot)\) \(\chi_{1309}(193,\cdot)\) \(\chi_{1309}(226,\cdot)\) \(\chi_{1309}(228,\cdot)\) \(\chi_{1309}(233,\cdot)\) \(\chi_{1309}(249,\cdot)\) \(\chi_{1309}(261,\cdot)\) \(\chi_{1309}(277,\cdot)\) \(\chi_{1309}(282,\cdot)\) \(\chi_{1309}(303,\cdot)\) \(\chi_{1309}(326,\cdot)\) \(\chi_{1309}(347,\cdot)\) \(\chi_{1309}(354,\cdot)\) \(\chi_{1309}(380,\cdot)\) \(\chi_{1309}(403,\cdot)\) \(\chi_{1309}(415,\cdot)\) \(\chi_{1309}(431,\cdot)\) \(\chi_{1309}(436,\cdot)\) \(\chi_{1309}(464,\cdot)\) \(\chi_{1309}(513,\cdot)\) \(\chi_{1309}(534,\cdot)\) \(\chi_{1309}(541,\cdot)\) ...

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: $\Q(\zeta_{240})$
Fixed field: Number field defined by a degree 240 polynomial (not computed)

Values on generators

\((1123,596,309)\) → \((e\left(\frac{2}{3}\right),e\left(\frac{9}{10}\right),e\left(\frac{5}{16}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(12\)\(13\)
\( \chi_{ 1309 }(39, a) \) \(1\)\(1\)\(e\left(\frac{73}{120}\right)\)\(e\left(\frac{43}{240}\right)\)\(e\left(\frac{13}{60}\right)\)\(e\left(\frac{119}{240}\right)\)\(e\left(\frac{63}{80}\right)\)\(e\left(\frac{33}{40}\right)\)\(e\left(\frac{43}{120}\right)\)\(e\left(\frac{5}{48}\right)\)\(e\left(\frac{19}{48}\right)\)\(e\left(\frac{3}{20}\right)\)
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)
 
\( \chi_{ 1309 }(39,a) \;\) at \(\;a = \) e.g. 2