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Properties

Label 1014.995
Modulus $1014$
Conductor $39$
Order $12$
Real no
Primitive no
Minimal no
Parity even

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1014, base_ring=CyclotomicField(12)) M = H._module chi = DirichletCharacter(H, M([6,11]))
 
Copy content pari:[g,chi] = znchar(Mod(995,1014))
 

Basic properties

Modulus: \(1014\)
Conductor: \(39\)
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 \(\chi_{39}(20,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 1014.k

\(\chi_{1014}(89,\cdot)\) \(\chi_{1014}(587,\cdot)\) \(\chi_{1014}(695,\cdot)\) \(\chi_{1014}(995,\cdot)\)

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: \(\Q(\zeta_{39})^+\)

Values on generators

\((677,847)\) → \((-1,e\left(\frac{11}{12}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)\(35\)
\( \chi_{ 1014 }(995, a) \) \(1\)\(1\)\(-i\)\(e\left(\frac{1}{12}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{7}{12}\right)\)\(e\left(\frac{2}{3}\right)\)\(-1\)\(e\left(\frac{1}{6}\right)\)\(i\)\(e\left(\frac{5}{6}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 1014 }(995,a) \;\) at \(\;a = \) e.g. 2