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Properties

Label 1309.1086
Modulus $1309$
Conductor $187$
Order $40$
Real no
Primitive no
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(40)) M = H._module chi = DirichletCharacter(H, M([0,12,15]))
 
Copy content gp:[g,chi] = znchar(Mod(1086, 1309))
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1309.1086");
 

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: \(187\)
Copy content comment:Conductor
 
Copy content sage:chi.conductor()
 
Copy content gp:znconreyconductor(g,chi)
 
Copy content magma:Conductor(chi);
 
Order: \(40\)
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: no, induced from \(\chi_{187}(151,\cdot)\)
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.cf

\(\chi_{1309}(8,\cdot)\) \(\chi_{1309}(127,\cdot)\) \(\chi_{1309}(134,\cdot)\) \(\chi_{1309}(162,\cdot)\) \(\chi_{1309}(281,\cdot)\) \(\chi_{1309}(365,\cdot)\) \(\chi_{1309}(393,\cdot)\) \(\chi_{1309}(491,\cdot)\) \(\chi_{1309}(512,\cdot)\) \(\chi_{1309}(519,\cdot)\) \(\chi_{1309}(722,\cdot)\) \(\chi_{1309}(750,\cdot)\) \(\chi_{1309}(876,\cdot)\) \(\chi_{1309}(1086,\cdot)\) \(\chi_{1309}(1107,\cdot)\) \(\chi_{1309}(1205,\cdot)\)

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: \(\Q(\zeta_{40})\)
Fixed field: 40.0.359624204259227998212313764863527746816862563620018205460931204658277030572367073.1

Values on generators

\((1123,596,309)\) → \((1,e\left(\frac{3}{10}\right),e\left(\frac{3}{8}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(12\)\(13\)
\( \chi_{ 1309 }(1086, a) \) \(-1\)\(1\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{31}{40}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{3}{40}\right)\)\(e\left(\frac{13}{40}\right)\)\(e\left(\frac{13}{20}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{4}{5}\right)\)
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)
 
\( \chi_{ 1309 }(1086,a) \;\) at \(\;a = \) e.g. 2