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Properties

Label 322.129
Modulus $322$
Conductor $161$
Order $66$
Real no
Primitive no
Minimal yes
Parity even

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(322, base_ring=CyclotomicField(66)) M = H._module chi = DirichletCharacter(H, M([11,63]))
 
Copy content pari:[g,chi] = znchar(Mod(129,322))
 

Basic properties

Modulus: \(322\)
Conductor: \(161\)
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: no, induced from \(\chi_{161}(129,\cdot)\)
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 322.o

\(\chi_{322}(5,\cdot)\) \(\chi_{322}(17,\cdot)\) \(\chi_{322}(19,\cdot)\) \(\chi_{322}(33,\cdot)\) \(\chi_{322}(61,\cdot)\) \(\chi_{322}(89,\cdot)\) \(\chi_{322}(103,\cdot)\) \(\chi_{322}(129,\cdot)\) \(\chi_{322}(143,\cdot)\) \(\chi_{322}(145,\cdot)\) \(\chi_{322}(157,\cdot)\) \(\chi_{322}(159,\cdot)\) \(\chi_{322}(171,\cdot)\) \(\chi_{322}(199,\cdot)\) \(\chi_{322}(201,\cdot)\) \(\chi_{322}(227,\cdot)\) \(\chi_{322}(241,\cdot)\) \(\chi_{322}(283,\cdot)\) \(\chi_{322}(297,\cdot)\) \(\chi_{322}(313,\cdot)\)

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

\((185,281)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{21}{22}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(9\)\(11\)\(13\)\(15\)\(17\)\(19\)\(25\)\(27\)
\( \chi_{ 322 }(129, a) \) \(1\)\(1\)\(e\left(\frac{29}{66}\right)\)\(e\left(\frac{26}{33}\right)\)\(e\left(\frac{29}{33}\right)\)\(e\left(\frac{17}{66}\right)\)\(e\left(\frac{19}{22}\right)\)\(e\left(\frac{5}{22}\right)\)\(e\left(\frac{28}{33}\right)\)\(e\left(\frac{5}{33}\right)\)\(e\left(\frac{19}{33}\right)\)\(e\left(\frac{7}{22}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 322 }(129,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 322 }(129,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 322 }(129,·),\chi_{ 322 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 322 }(129,·)) \;\) at \(\; a,b = \) e.g. 1,2