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Properties

Label 3751.1721
Modulus $3751$
Conductor $341$
Order $5$
Real no
Primitive no
Minimal no
Parity even

Related objects

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Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3751, base_ring=CyclotomicField(10)) M = H._module chi = DirichletCharacter(H, M([4,2]))
 
Copy content pari:[g,chi] = znchar(Mod(1721,3751))
 

Basic properties

Modulus: \(3751\)
Conductor: \(341\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(5\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{341}(16,\cdot)\)
Copy content sage:chi.is_primitive()
 
Copy content pari:#znconreyconductor(g,chi)==1
 
Minimal: no
Parity: even
Copy content sage:chi.is_odd()
 
Copy content pari:zncharisodd(g,chi)
 

Galois orbit 3751.f

\(\chi_{3751}(686,\cdot)\) \(\chi_{3751}(1721,\cdot)\) \(\chi_{3751}(2302,\cdot)\) \(\chi_{3751}(2792,\cdot)\)

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: 5.5.13521270961.3

Values on generators

\((2543,2421)\) → \((e\left(\frac{2}{5}\right),e\left(\frac{1}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(2\)\(3\)\(4\)\(5\)\(6\)\(7\)\(8\)\(9\)\(10\)\(12\)
\( \chi_{ 3751 }(1721, a) \) \(1\)\(1\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{2}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{4}{5}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 3751 }(1721,a) \;\) at \(\;a = \) e.g. 2