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Properties

Label 6660.71
Modulus $6660$
Conductor $444$
Order $18$
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(6660, base_ring=CyclotomicField(18)) M = H._module chi = DirichletCharacter(H, M([9,9,0,4]))
 
Copy content pari:[g,chi] = znchar(Mod(71,6660))
 

Basic properties

Modulus: \(6660\)
Conductor: \(444\)
Copy content sage:chi.conductor()
 
Copy content pari:znconreyconductor(g,chi)
 
Order: \(18\)
Copy content sage:chi.multiplicative_order()
 
Copy content pari:charorder(g,chi)
 
Real: no
Primitive: no, induced from \(\chi_{444}(71,\cdot)\)
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 6660.if

\(\chi_{6660}(71,\cdot)\) \(\chi_{6660}(971,\cdot)\) \(\chi_{6660}(2051,\cdot)\) \(\chi_{6660}(2771,\cdot)\) \(\chi_{6660}(6191,\cdot)\) \(\chi_{6660}(6371,\cdot)\)

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: Number field defined by a degree 18 polynomial

Values on generators

\((3331,3701,3997,3961)\) → \((-1,-1,1,e\left(\frac{2}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(7\)\(11\)\(13\)\(17\)\(19\)\(23\)\(29\)\(31\)\(41\)\(43\)
\( \chi_{ 6660 }(71, a) \) \(1\)\(1\)\(e\left(\frac{11}{18}\right)\)\(e\left(\frac{2}{3}\right)\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{6}\right)\)\(-1\)\(e\left(\frac{17}{18}\right)\)\(-1\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 6660 }(71,a) \;\) at \(\;a = \) e.g. 2