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Properties

Label 600.409
Modulus $600$
Conductor $25$
Order $10$
Real no
Primitive no
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(600, base_ring=CyclotomicField(10)) M = H._module chi = DirichletCharacter(H, M([0,0,0,7]))
 
Copy content pari:[g,chi] = znchar(Mod(409,600))
 

Basic properties

Modulus: \(600\)
Conductor: \(25\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(10\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{25}(9,\cdot)\)
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 600.bc

\(\chi_{600}(169,\cdot)\) \(\chi_{600}(289,\cdot)\) \(\chi_{600}(409,\cdot)\) \(\chi_{600}(529,\cdot)\)

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: \(\Q(\zeta_{5})\)
Fixed field: \(\Q(\zeta_{25})^+\)

Values on generators

\((151,301,401,577)\) → \((1,1,1,e\left(\frac{7}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 600 }(409, a) \) \(1\)\(1\)\(-1\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{1}{10}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{4}{5}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 600 }(409,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 600 }(409,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 600 }(409,·),\chi_{ 600 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 600 }(409,·)) \;\) at \(\; a,b = \) e.g. 1,2