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Properties

Label 8550.49
Modulus $8550$
Conductor $855$
Order $6$
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(8550, base_ring=CyclotomicField(6)) M = H._module chi = DirichletCharacter(H, M([2,3,4]))
 
Copy content pari:[g,chi] = znchar(Mod(49,8550))
 

Basic properties

Modulus: \(8550\)
Conductor: \(855\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(6\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{855}(49,\cdot)\)
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 8550.be

\(\chi_{8550}(49,\cdot)\) \(\chi_{8550}(349,\cdot)\)

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: \(\mathbb{Q}(\zeta_3)\)
Fixed field: 6.6.106879510125.4

Values on generators

\((1901,1027,1351)\) → \((e\left(\frac{1}{3}\right),-1,e\left(\frac{2}{3}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 8550 }(49, a) \) \(1\)\(1\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{3}\right)\)\(-1\)\(e\left(\frac{1}{6}\right)\)\(-1\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{2}{3}\right)\)\(-1\)\(e\left(\frac{1}{3}\right)\)\(-1\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 8550 }(49,a) \;\) at \(\;a = \) e.g. 2