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Properties

Label 1710.871
Modulus $1710$
Conductor $171$
Order $9$
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(1710, base_ring=CyclotomicField(18)) M = H._module chi = DirichletCharacter(H, M([12,0,4]))
 
Copy content pari:[g,chi] = znchar(Mod(871,1710))
 

Basic properties

Modulus: \(1710\)
Conductor: \(171\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(9\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{171}(16,\cdot)\)
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 1710.bu

\(\chi_{1710}(481,\cdot)\) \(\chi_{1710}(511,\cdot)\) \(\chi_{1710}(871,\cdot)\) \(\chi_{1710}(1111,\cdot)\) \(\chi_{1710}(1201,\cdot)\) \(\chi_{1710}(1411,\cdot)\)

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: \(\Q(\zeta_{9})\)
Fixed field: 9.9.9025761726072081.2

Values on generators

\((191,1027,1351)\) → \((e\left(\frac{2}{3}\right),1,e\left(\frac{2}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(23\)\(29\)\(31\)\(37\)\(41\)\(43\)
\( \chi_{ 1710 }(871, a) \) \(1\)\(1\)\(1\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{2}{3}\right)\)\(1\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{2}{9}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 1710 }(871,a) \;\) at \(\;a = \) e.g. 2