↑

Properties

Label 608.81
Modulus $608$
Conductor $152$
Order $18$
Real no
Primitive no
Minimal no
Parity even

Related objects

Downloads

Learn more

Show commands: Pari/GP / SageMath
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(608, base_ring=CyclotomicField(18)) M = H._module chi = DirichletCharacter(H, M([0,9,16]))
 
Copy content pari:[g,chi] = znchar(Mod(81,608))
 

Basic properties

Modulus: \(608\)
Conductor: \(152\)
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 \(\chi_{152}(5,\cdot)\)
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 608.bf

\(\chi_{608}(17,\cdot)\) \(\chi_{608}(81,\cdot)\) \(\chi_{608}(177,\cdot)\) \(\chi_{608}(465,\cdot)\) \(\chi_{608}(529,\cdot)\) \(\chi_{608}(593,\cdot)\)

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: 18.18.38713951190154487490850848768.1

Values on generators

\((191,229,97)\) → \((1,-1,e\left(\frac{8}{9}\right))\)

First values

\(a\) \(-1\)\(1\)\(3\)\(5\)\(7\)\(9\)\(11\)\(13\)\(15\)\(17\)\(21\)\(23\)
\( \chi_{ 608 }(81, a) \) \(1\)\(1\)\(e\left(\frac{1}{18}\right)\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{7}{9}\right)\)
Copy content sage:chi.jacobi_sum(n)
 
\( \chi_{ 608 }(81,a) \;\) at \(\;a = \) e.g. 2

Gauss sum

Copy content sage:chi.gauss_sum(a)
 
Copy content pari:znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 608 }(81,·) )\;\) at \(\;a = \) e.g. 2

Jacobi sum

Copy content sage:chi.jacobi_sum(n)
 
\( J(\chi_{ 608 }(81,·),\chi_{ 608 }(n,·)) \;\) for \( \; n = \) e.g. 1

Kloosterman sum

Copy content sage:chi.kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 608 }(81,·)) \;\) at \(\; a,b = \) e.g. 1,2