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Properties

Label 1309.773
Modulus $1309$
Conductor $1309$
Order $120$
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(1309, base_ring=CyclotomicField(120)) M = H._module chi = DirichletCharacter(H, M([20,96,75]))
 
Copy content gp:[g,chi] = znchar(Mod(773, 1309))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1309.773");
 

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: \(120\)
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 1309.cv

\(\chi_{1309}(26,\cdot)\) \(\chi_{1309}(59,\cdot)\) \(\chi_{1309}(185,\cdot)\) \(\chi_{1309}(213,\cdot)\) \(\chi_{1309}(229,\cdot)\) \(\chi_{1309}(236,\cdot)\) \(\chi_{1309}(257,\cdot)\) \(\chi_{1309}(355,\cdot)\) \(\chi_{1309}(383,\cdot)\) \(\chi_{1309}(416,\cdot)\) \(\chi_{1309}(423,\cdot)\) \(\chi_{1309}(444,\cdot)\) \(\chi_{1309}(467,\cdot)\) \(\chi_{1309}(542,\cdot)\) \(\chi_{1309}(570,\cdot)\) \(\chi_{1309}(586,\cdot)\) \(\chi_{1309}(614,\cdot)\) \(\chi_{1309}(621,\cdot)\) \(\chi_{1309}(654,\cdot)\) \(\chi_{1309}(740,\cdot)\) \(\chi_{1309}(773,\cdot)\) \(\chi_{1309}(801,\cdot)\) \(\chi_{1309}(808,\cdot)\) \(\chi_{1309}(852,\cdot)\) \(\chi_{1309}(927,\cdot)\) \(\chi_{1309}(950,\cdot)\) \(\chi_{1309}(971,\cdot)\) \(\chi_{1309}(1039,\cdot)\) \(\chi_{1309}(1137,\cdot)\) \(\chi_{1309}(1158,\cdot)\) ...

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: $\Q(\zeta_{120})$
Fixed field: Number field defined by a degree 120 polynomial (not computed)

Values on generators

\((1123,596,309)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{4}{5}\right),e\left(\frac{5}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(12\)\(13\)
\( \chi_{ 1309 }(773, a) \) \(-1\)\(1\)\(e\left(\frac{53}{60}\right)\)\(e\left(\frac{23}{120}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{19}{120}\right)\)\(e\left(\frac{3}{40}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{23}{60}\right)\)\(e\left(\frac{1}{24}\right)\)\(e\left(\frac{23}{24}\right)\)\(e\left(\frac{4}{5}\right)\)
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)
 
\( \chi_{ 1309 }(773,a) \;\) at \(\;a = \) e.g. 2