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Properties

Label 648.301
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([0,27,38]))
 
Copy content pari:[g,chi] = znchar(Mod(301,648))
 

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.bd

\(\chi_{648}(13,\cdot)\) \(\chi_{648}(61,\cdot)\) \(\chi_{648}(85,\cdot)\) \(\chi_{648}(133,\cdot)\) \(\chi_{648}(157,\cdot)\) \(\chi_{648}(205,\cdot)\) \(\chi_{648}(229,\cdot)\) \(\chi_{648}(277,\cdot)\) \(\chi_{648}(301,\cdot)\) \(\chi_{648}(349,\cdot)\) \(\chi_{648}(373,\cdot)\) \(\chi_{648}(421,\cdot)\) \(\chi_{648}(445,\cdot)\) \(\chi_{648}(493,\cdot)\) \(\chi_{648}(517,\cdot)\) \(\chi_{648}(565,\cdot)\) \(\chi_{648}(589,\cdot)\) \(\chi_{648}(637,\cdot)\)

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

\((487,325,569)\) → \((1,-1,e\left(\frac{19}{27}\right))\)

First values

\(a\) \(-1\)\(1\)\(5\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(25\)\(29\)\(31\)
\( \chi_{ 648 }(301, a) \) \(1\)\(1\)\(e\left(\frac{37}{54}\right)\)\(e\left(\frac{7}{27}\right)\)\(e\left(\frac{35}{54}\right)\)\(e\left(\frac{7}{54}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{20}{27}\right)\)\(e\left(\frac{10}{27}\right)\)\(e\left(\frac{29}{54}\right)\)\(e\left(\frac{2}{27}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 648 }(301,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 648 }(301,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 648 }(301,·),\chi_{ 648 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 648 }(301,·)) \;\) at \(\; a,b = \) e.g. 1,2