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Properties

Label 3000.149
Modulus $3000$
Conductor $600$
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(3000, base_ring=CyclotomicField(10)) M = H._module chi = DirichletCharacter(H, M([0,5,5,1]))
 
Copy content pari:[g,chi] = znchar(Mod(149,3000))
 

Basic properties

Modulus: \(3000\)
Conductor: \(600\)
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 \(\chi_{600}(29,\cdot)\)
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 3000.z

\(\chi_{3000}(149,\cdot)\) \(\chi_{3000}(1349,\cdot)\) \(\chi_{3000}(1949,\cdot)\) \(\chi_{3000}(2549,\cdot)\)

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: 10.0.6075000000000000000.26

Values on generators

\((751,1501,1001,2377)\) → \((1,-1,-1,e\left(\frac{1}{10}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 3000 }(149, a) \) \(-1\)\(1\)\(-1\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{3}{10}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{9}{10}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 3000 }(149,a) \;\) at \(\;a = \) e.g. 2