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Properties

Label 26.15
Modulus $26$
Conductor $13$
Order $12$
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(26, base_ring=CyclotomicField(12)) M = H._module chi = DirichletCharacter(H, M([1]))
 
Copy content pari:[g,chi] = znchar(Mod(15,26))
 

Basic properties

Modulus: \(26\)
Conductor: \(13\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(12\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{13}(2,\cdot)\)
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 26.f

\(\chi_{26}(7,\cdot)\) \(\chi_{26}(11,\cdot)\) \(\chi_{26}(15,\cdot)\) \(\chi_{26}(19,\cdot)\)

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: \(\Q(\zeta_{12})\)
Fixed field: \(\Q(\zeta_{13})\)

Values on generators

\(15\) → \(e\left(\frac{1}{12}\right)\)

Values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(15\)\(17\)\(19\)\(21\)\(23\)
\( \chi_{ 26 }(15, a) \) \(-1\)\(1\)\(e\left(\frac{1}{3}\right)\)\(-i\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{5}{12}\right)\)\(i\)\(e\left(\frac{5}{6}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 26 }(15,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 26 }(15,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 26 }(15,·),\chi_{ 26 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 26 }(15,·)) \;\) at \(\; a,b = \) e.g. 1,2