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Properties

Label 1309.9
Modulus $1309$
Conductor $1309$
Order $120$
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(120)) M = H._module chi = DirichletCharacter(H, M([40,72,15]))
 
Copy content gp:[g,chi] = znchar(Mod(9, 1309))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1309.9");
 

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: 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.cx

\(\chi_{1309}(9,\cdot)\) \(\chi_{1309}(25,\cdot)\) \(\chi_{1309}(53,\cdot)\) \(\chi_{1309}(60,\cdot)\) \(\chi_{1309}(93,\cdot)\) \(\chi_{1309}(179,\cdot)\) \(\chi_{1309}(212,\cdot)\) \(\chi_{1309}(240,\cdot)\) \(\chi_{1309}(247,\cdot)\) \(\chi_{1309}(291,\cdot)\) \(\chi_{1309}(366,\cdot)\) \(\chi_{1309}(389,\cdot)\) \(\chi_{1309}(410,\cdot)\) \(\chi_{1309}(478,\cdot)\) \(\chi_{1309}(576,\cdot)\) \(\chi_{1309}(597,\cdot)\) \(\chi_{1309}(620,\cdot)\) \(\chi_{1309}(746,\cdot)\) \(\chi_{1309}(774,\cdot)\) \(\chi_{1309}(807,\cdot)\) \(\chi_{1309}(933,\cdot)\) \(\chi_{1309}(961,\cdot)\) \(\chi_{1309}(977,\cdot)\) \(\chi_{1309}(984,\cdot)\) \(\chi_{1309}(1005,\cdot)\) \(\chi_{1309}(1103,\cdot)\) \(\chi_{1309}(1131,\cdot)\) \(\chi_{1309}(1164,\cdot)\) \(\chi_{1309}(1171,\cdot)\) \(\chi_{1309}(1192,\cdot)\) ...

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

\((1123,596,309)\) → \((e\left(\frac{1}{3}\right),e\left(\frac{3}{5}\right),e\left(\frac{1}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(12\)\(13\)
\( \chi_{ 1309 }(9, a) \) \(1\)\(1\)\(e\left(\frac{1}{60}\right)\)\(e\left(\frac{31}{120}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{83}{120}\right)\)\(e\left(\frac{11}{40}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{31}{60}\right)\)\(e\left(\frac{17}{24}\right)\)\(e\left(\frac{7}{24}\right)\)\(e\left(\frac{1}{10}\right)\)
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)
 
\( \chi_{ 1309 }(9,a) \;\) at \(\;a = \) e.g. 2