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Properties

Label 385.47
Modulus $385$
Conductor $385$
Order $60$
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(385, base_ring=CyclotomicField(60)) M = H._module chi = DirichletCharacter(H, M([15,50,48]))
 
Copy content pari:[g,chi] = znchar(Mod(47,385))
 

Basic properties

Modulus: \(385\)
Conductor: \(385\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(60\)
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 385.bu

\(\chi_{385}(3,\cdot)\) \(\chi_{385}(38,\cdot)\) \(\chi_{385}(47,\cdot)\) \(\chi_{385}(82,\cdot)\) \(\chi_{385}(103,\cdot)\) \(\chi_{385}(108,\cdot)\) \(\chi_{385}(152,\cdot)\) \(\chi_{385}(157,\cdot)\) \(\chi_{385}(192,\cdot)\) \(\chi_{385}(213,\cdot)\) \(\chi_{385}(257,\cdot)\) \(\chi_{385}(262,\cdot)\) \(\chi_{385}(278,\cdot)\) \(\chi_{385}(313,\cdot)\) \(\chi_{385}(367,\cdot)\) \(\chi_{385}(383,\cdot)\)

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: \(\Q(\zeta_{60})\)
Fixed field: Number field defined by a degree 60 polynomial

Values on generators

\((232,276,211)\) → \((i,e\left(\frac{5}{6}\right),e\left(\frac{4}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(6\)\(8\)\(9\)\(12\)\(13\)\(16\)\(17\)
\( \chi_{ 385 }(47, a) \) \(1\)\(1\)\(e\left(\frac{43}{60}\right)\)\(e\left(\frac{59}{60}\right)\)\(e\left(\frac{13}{30}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{29}{30}\right)\)\(e\left(\frac{5}{12}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{13}{15}\right)\)\(e\left(\frac{17}{60}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 385 }(47,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 385 }(47,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 385 }(47,·),\chi_{ 385 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 385 }(47,·)) \;\) at \(\; a,b = \) e.g. 1,2