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Properties

Label 3660.1681
Modulus $3660$
Conductor $61$
Order $5$
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(3660, base_ring=CyclotomicField(10)) M = H._module chi = DirichletCharacter(H, M([0,0,0,8]))
 
Copy content pari:[g,chi] = znchar(Mod(1681,3660))
 

Basic properties

Modulus: \(3660\)
Conductor: \(61\)
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 \(\chi_{61}(34,\cdot)\)
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 3660.bp

\(\chi_{3660}(241,\cdot)\) \(\chi_{3660}(1681,\cdot)\) \(\chi_{3660}(2521,\cdot)\) \(\chi_{3660}(3181,\cdot)\)

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: 5.5.13845841.1

Values on generators

\((1831,2441,2197,3601)\) → \((1,1,1,e\left(\frac{4}{5}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(37\)\(41\)
\( \chi_{ 3660 }(1681, a) \) \(1\)\(1\)\(e\left(\frac{1}{5}\right)\)\(1\)\(1\)\(e\left(\frac{3}{5}\right)\)\(e\left(\frac{4}{5}\right)\)\(e\left(\frac{3}{5}\right)\)\(1\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{1}{5}\right)\)\(e\left(\frac{1}{5}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 3660 }(1681,a) \;\) at \(\;a = \) e.g. 2