Properties

Conductor 63
Order 6
Real no
Primitive no
Minimal yes
Parity even
Orbit label 252.bm

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(252)
 
sage: chi = H[173]
 
pari: [g,chi] = znchar(Mod(173,252))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 63
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 6
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 = 252.bm
Orbit index = 39

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 ]
 

\(\chi_{252}(173,\cdot)\) \(\chi_{252}(185,\cdot)\)

Values on generators

\((127,29,73)\) → \((1,e\left(\frac{1}{6}\right),e\left(\frac{5}{6}\right))\)

Values

-115111317192325293137
\(1\)\(1\)\(1\)\(-1\)\(e\left(\frac{5}{6}\right)\)\(e\left(\frac{1}{3}\right)\)\(e\left(\frac{1}{6}\right)\)\(-1\)\(1\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{2}{3}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{3})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 252 }(173,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{252}(173,\cdot)) = \sum_{r\in \Z/252\Z} \chi_{252}(173,r) e\left(\frac{r}{126}\right) = 11.6593145088+10.7731325613i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 252 }(173,·),\chi_{ 252 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{252}(173,\cdot),\chi_{252}(1,\cdot)) = \sum_{r\in \Z/252\Z} \chi_{252}(173,r) \chi_{252}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 252 }(173,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{252}(173,·)) = \sum_{r \in \Z/252\Z} \chi_{252}(173,r) e\left(\frac{1 r + 2 r^{-1}}{252}\right) = -0.0 \)