Properties

Conductor 189
Order 9
Real no
Primitive no
Minimal yes
Parity even
Orbit label 378.w

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(378)
 
sage: chi = H[25]
 
pari: [g,chi] = znchar(Mod(25,378))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 189
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 9
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 = 378.w
Orbit index = 23

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 ]
 

\(\chi_{378}(25,\cdot)\) \(\chi_{378}(121,\cdot)\) \(\chi_{378}(151,\cdot)\) \(\chi_{378}(247,\cdot)\) \(\chi_{378}(277,\cdot)\) \(\chi_{378}(373,\cdot)\)

Values on generators

\((29,325)\) → \((e\left(\frac{5}{9}\right),e\left(\frac{2}{3}\right))\)

Values

-115111317192325293137
\(1\)\(1\)\(e\left(\frac{1}{9}\right)\)\(e\left(\frac{8}{9}\right)\)\(e\left(\frac{4}{9}\right)\)\(1\)\(1\)\(e\left(\frac{4}{9}\right)\)\(e\left(\frac{2}{9}\right)\)\(e\left(\frac{5}{9}\right)\)\(e\left(\frac{7}{9}\right)\)\(e\left(\frac{2}{3}\right)\)
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)
 
\( \tau_{ a }( \chi_{ 378 }(25,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{378}(25,\cdot)) = \sum_{r\in \Z/378\Z} \chi_{378}(25,r) e\left(\frac{r}{189}\right) = 7.4946982186+-11.5251680514i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 378 }(25,·),\chi_{ 378 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{378}(25,\cdot),\chi_{378}(1,\cdot)) = \sum_{r\in \Z/378\Z} \chi_{378}(25,r) \chi_{378}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 378 }(25,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{378}(25,·)) = \sum_{r \in \Z/378\Z} \chi_{378}(25,r) e\left(\frac{1 r + 2 r^{-1}}{378}\right) = -11.4755853081+4.1767714729i \)