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Properties

Label 67.11
Modulus $67$
Conductor $67$
Order $66$
Real no
Primitive yes
Minimal yes
Parity odd

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(67, base_ring=CyclotomicField(66)) M = H._module chi = DirichletCharacter(H, M([59]))
 
Copy content pari:[g,chi] = znchar(Mod(11,67))
 

Basic properties

Modulus: \(67\)
Conductor: \(67\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(66\)
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: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 67.h

\(\chi_{67}(2,\cdot)\) \(\chi_{67}(7,\cdot)\) \(\chi_{67}(11,\cdot)\) \(\chi_{67}(12,\cdot)\) \(\chi_{67}(13,\cdot)\) \(\chi_{67}(18,\cdot)\) \(\chi_{67}(20,\cdot)\) \(\chi_{67}(28,\cdot)\) \(\chi_{67}(31,\cdot)\) \(\chi_{67}(32,\cdot)\) \(\chi_{67}(34,\cdot)\) \(\chi_{67}(41,\cdot)\) \(\chi_{67}(44,\cdot)\) \(\chi_{67}(46,\cdot)\) \(\chi_{67}(48,\cdot)\) \(\chi_{67}(50,\cdot)\) \(\chi_{67}(51,\cdot)\) \(\chi_{67}(57,\cdot)\) \(\chi_{67}(61,\cdot)\) \(\chi_{67}(63,\cdot)\)

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: \(\Q(\zeta_{33})\)
Fixed field: Number field defined by a degree 66 polynomial

Values on generators

\(2\) → \(e\left(\frac{59}{66}\right)\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(11\)
\( \chi_{ 67 }(11, a) \) \(-1\)\(1\)\(e\left(\frac{59}{66}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{26}{33}\right)\)\(e\left(\frac{9}{22}\right)\)\(e\left(\frac{25}{33}\right)\)\(e\left(\frac{37}{66}\right)\)\(e\left(\frac{15}{22}\right)\)\(e\left(\frac{8}{11}\right)\)\(e\left(\frac{10}{33}\right)\)\(e\left(\frac{49}{66}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 67 }(11,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 67 }(11,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 67 }(11,·),\chi_{ 67 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 67 }(11,·)) \;\) at \(\; a,b = \) e.g. 1,2