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Properties

Label 1309.43
Modulus $1309$
Conductor $187$
Order $8$
Real no
Primitive no
Minimal yes
Parity odd

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1309, base_ring=CyclotomicField(8)) M = H._module chi = DirichletCharacter(H, M([0,4,1]))
 
Copy content pari:[g,chi] = znchar(Mod(43,1309))
 

Basic properties

Modulus: \(1309\)
Conductor: \(187\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(8\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{187}(43,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 1309.x

\(\chi_{1309}(43,\cdot)\) \(\chi_{1309}(274,\cdot)\) \(\chi_{1309}(967,\cdot)\) \(\chi_{1309}(1198,\cdot)\)

Copy content sage:chi.galois_orbit()
 
Copy content pari:order = charorder(g,chi) [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

Related number fields

Field of values: \(\Q(\zeta_{8})\)
Fixed field: 8.0.6007768511393.1

Values on generators

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

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(8\)\(9\)\(10\)\(12\)\(13\)
\( \chi_{ 1309 }(43, a) \) \(-1\)\(1\)\(i\)\(e\left(\frac{1}{8}\right)\)\(-1\)\(e\left(\frac{5}{8}\right)\)\(e\left(\frac{3}{8}\right)\)\(-i\)\(i\)\(e\left(\frac{7}{8}\right)\)\(e\left(\frac{5}{8}\right)\)\(1\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 1309 }(43,a) \;\) at \(\;a = \) e.g. 2