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Properties

Label 648.299
Modulus $648$
Conductor $648$
Order $54$
Real no
Primitive yes
Minimal yes
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(54)) M = H._module chi = DirichletCharacter(H, M([27,27,19]))
 
Copy content pari:[g,chi] = znchar(Mod(299,648))
 

Basic properties

Modulus: \(648\)
Conductor: \(648\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(54\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: yes
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: yes
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 648.bb

\(\chi_{648}(11,\cdot)\) \(\chi_{648}(59,\cdot)\) \(\chi_{648}(83,\cdot)\) \(\chi_{648}(131,\cdot)\) \(\chi_{648}(155,\cdot)\) \(\chi_{648}(203,\cdot)\) \(\chi_{648}(227,\cdot)\) \(\chi_{648}(275,\cdot)\) \(\chi_{648}(299,\cdot)\) \(\chi_{648}(347,\cdot)\) \(\chi_{648}(371,\cdot)\) \(\chi_{648}(419,\cdot)\) \(\chi_{648}(443,\cdot)\) \(\chi_{648}(491,\cdot)\) \(\chi_{648}(515,\cdot)\) \(\chi_{648}(563,\cdot)\) \(\chi_{648}(587,\cdot)\) \(\chi_{648}(635,\cdot)\)

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: \(\Q(\zeta_{27})\)
Fixed field: Number field defined by a degree 54 polynomial

Values on generators

\((487,325,569)\) → \((-1,-1,e\left(\frac{19}{54}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 648 }(299, a) \) \(1\)\(1\)\(e\left(\frac{16}{27}\right)\)\(e\left(\frac{7}{54}\right)\)\(e\left(\frac{31}{54}\right)\)\(e\left(\frac{17}{54}\right)\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{10}{27}\right)\)\(e\left(\frac{5}{27}\right)\)\(e\left(\frac{14}{27}\right)\)\(e\left(\frac{29}{54}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 648 }(299,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 648 }(299,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 648 }(299,·),\chi_{ 648 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 648 }(299,·)) \;\) at \(\; a,b = \) e.g. 1,2