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Properties

Label 2000.1151
Modulus $2000$
Conductor $100$
Order $10$
Real no
Primitive no
Minimal no
Parity odd

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2000, base_ring=CyclotomicField(10)) M = H._module chi = DirichletCharacter(H, M([5,0,4]))
 
Copy content pari:[g,chi] = znchar(Mod(1151,2000))
 

Basic properties

Modulus: \(2000\)
Conductor: \(100\)
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 \(\chi_{100}(31,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: no
Parity: odd
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 2000.z

\(\chi_{2000}(351,\cdot)\) \(\chi_{2000}(1151,\cdot)\) \(\chi_{2000}(1551,\cdot)\) \(\chi_{2000}(1951,\cdot)\)

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: 10.0.156250000000000.1

Values on generators

\((751,501,1377)\) → \((-1,1,e\left(\frac{2}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(7\)\(9\)\(11\)\(13\)\(17\)\(19\)\(21\)\(23\)\(27\)
\( \chi_{ 2000 }(1151, a) \) \(-1\)\(1\)\(e\left(\frac{3}{10}\right)\)\(-1\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{9}{10}\right)\)\(e\left(\frac{9}{10}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 2000 }(1151,a) \;\) at \(\;a = \) e.g. 2