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Properties

Label 225.47
Modulus $225$
Conductor $225$
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(225, base_ring=CyclotomicField(60)) M = H._module chi = DirichletCharacter(H, M([10,51]))
 
Copy content pari:[g,chi] = znchar(Mod(47,225))
 

Basic properties

Modulus: \(225\)
Conductor: \(225\)
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 225.w

\(\chi_{225}(2,\cdot)\) \(\chi_{225}(23,\cdot)\) \(\chi_{225}(38,\cdot)\) \(\chi_{225}(47,\cdot)\) \(\chi_{225}(77,\cdot)\) \(\chi_{225}(83,\cdot)\) \(\chi_{225}(92,\cdot)\) \(\chi_{225}(113,\cdot)\) \(\chi_{225}(122,\cdot)\) \(\chi_{225}(128,\cdot)\) \(\chi_{225}(137,\cdot)\) \(\chi_{225}(158,\cdot)\) \(\chi_{225}(167,\cdot)\) \(\chi_{225}(173,\cdot)\) \(\chi_{225}(203,\cdot)\) \(\chi_{225}(212,\cdot)\)

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

\((101,127)\) → \((e\left(\frac{1}{6}\right),e\left(\frac{17}{20}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(4\)\(7\)\(8\)\(11\)\(13\)\(14\)\(16\)\(17\)\(19\)
\( \chi_{ 225 }(47, a) \) \(1\)\(1\)\(e\left(\frac{1}{60}\right)\)\(e\left(\frac{1}{30}\right)\)\(e\left(\frac{11}{12}\right)\)\(e\left(\frac{1}{20}\right)\)\(e\left(\frac{23}{30}\right)\)\(e\left(\frac{29}{60}\right)\)\(e\left(\frac{14}{15}\right)\)\(e\left(\frac{1}{15}\right)\)\(e\left(\frac{11}{20}\right)\)\(e\left(\frac{3}{10}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 225 }(47,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 225 }(47,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 225 }(47,·),\chi_{ 225 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 225 }(47,·)) \;\) at \(\; a,b = \) e.g. 1,2