Properties

Conductor 189
Order 18
Real no
Primitive no
Minimal yes
Parity even
Orbit label 756.ck

Related objects

Learn more about

Show commands for: SageMath / Pari/GP
sage: from dirichlet_conrey import DirichletGroup_conrey # requires nonstandard Sage package to be installed
 
sage: H = DirichletGroup_conrey(756)
 
sage: chi = H[5]
 
pari: [g,chi] = znchar(Mod(5,756))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 189
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 18
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = no
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = even
Orbit label = 756.ck
Orbit index = 63

Galois orbit

sage: chi.sage_character().galois_orbit()
 
pari: order = charorder(g,chi)
 
pari: [ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
 

\(\chi_{756}(5,\cdot)\) \(\chi_{756}(101,\cdot)\) \(\chi_{756}(257,\cdot)\) \(\chi_{756}(353,\cdot)\) \(\chi_{756}(509,\cdot)\) \(\chi_{756}(605,\cdot)\)

Values on generators

\((379,29,325)\) → \((1,e\left(\frac{5}{18}\right),e\left(\frac{5}{6}\right))\)

Values

-115111317192325293137
\(1\)\(1\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{17}{18}\right)\)\(e\left(\frac{13}{18}\right)\)\(1\)\(-1\)\(e\left(\frac{13}{18}\right)\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{5}{18}\right)\)\(e\left(\frac{7}{18}\right)\)\(e\left(\frac{1}{3}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{9})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 756 }(5,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{756}(5,\cdot)) = \sum_{r\in \Z/756\Z} \chi_{756}(5,r) e\left(\frac{r}{378}\right) = 1.8839619149+27.4308346119i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 756 }(5,·),\chi_{ 756 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{756}(5,\cdot),\chi_{756}(1,\cdot)) = \sum_{r\in \Z/756\Z} \chi_{756}(5,r) \chi_{756}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 756 }(5,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{756}(5,·)) = \sum_{r \in \Z/756\Z} \chi_{756}(5,r) e\left(\frac{1 r + 2 r^{-1}}{756}\right) = -0.0 \)