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Properties

Label 648.397
Modulus $648$
Conductor $216$
Order $18$
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(648, base_ring=CyclotomicField(18)) M = H._module chi = DirichletCharacter(H, M([0,9,10]))
 
Copy content pari:[g,chi] = znchar(Mod(397,648))
 

Basic properties

Modulus: \(648\)
Conductor: \(216\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(18\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{216}(133,\cdot)\)
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 648.t

\(\chi_{648}(37,\cdot)\) \(\chi_{648}(181,\cdot)\) \(\chi_{648}(253,\cdot)\) \(\chi_{648}(397,\cdot)\) \(\chi_{648}(469,\cdot)\) \(\chi_{648}(613,\cdot)\)

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: \(\Q(\zeta_{9})\)
Fixed field: 18.18.132173713091594538512566714368.1

Values on generators

\((487,325,569)\) → \((1,-1,e\left(\frac{5}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 648 }(397, a) \) \(1\)\(1\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{1}{9}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 648 }(397,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 648 }(397,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 648 }(397,·),\chi_{ 648 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 648 }(397,·)) \;\) at \(\; a,b = \) e.g. 1,2