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Properties

Label 1078.67
Modulus $1078$
Conductor $7$
Order $3$
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(1078, base_ring=CyclotomicField(6)) M = H._module chi = DirichletCharacter(H, M([4,0]))
 
Copy content pari:[g,chi] = znchar(Mod(67,1078))
 

Basic properties

Modulus: \(1078\)
Conductor: \(7\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(3\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{7}(4,\cdot)\)
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 1078.e

\(\chi_{1078}(67,\cdot)\) \(\chi_{1078}(177,\cdot)\)

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: \(\mathbb{Q}(\zeta_3)\)
Fixed field: \(\Q(\zeta_{7})^+\)

Values on generators

\((199,981)\) → \((e\left(\frac{2}{3}\right),1)\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(13\)\(15\)\(17\)\(19\)\(23\)\(25\)\(27\)
\( \chi_{ 1078 }(67, a) \) \(1\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(1\)\(1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(1\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 1078 }(67,a) \;\) at \(\;a = \) e.g. 2