Properties

Conductor 175
Order 60
Real no
Primitive no
Minimal yes
Parity even
Orbit label 350.x

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(350)
 
sage: chi = H[3]
 
pari: [g,chi] = znchar(Mod(3,350))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 175
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 60
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 = 350.x
Orbit index = 24

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 ]
 

\(\chi_{350}(3,\cdot)\) \(\chi_{350}(17,\cdot)\) \(\chi_{350}(33,\cdot)\) \(\chi_{350}(47,\cdot)\) \(\chi_{350}(73,\cdot)\) \(\chi_{350}(87,\cdot)\) \(\chi_{350}(103,\cdot)\) \(\chi_{350}(117,\cdot)\) \(\chi_{350}(173,\cdot)\) \(\chi_{350}(187,\cdot)\) \(\chi_{350}(213,\cdot)\) \(\chi_{350}(227,\cdot)\) \(\chi_{350}(283,\cdot)\) \(\chi_{350}(297,\cdot)\) \(\chi_{350}(313,\cdot)\) \(\chi_{350}(327,\cdot)\)

Values on generators

\((127,101)\) → \((e\left(\frac{7}{20}\right),e\left(\frac{1}{6}\right))\)

Values

-11391113171923272931
\(1\)\(1\)\(e\left(\frac{37}{60}\right)\)\(e\left(\frac{7}{30}\right)\)\(e\left(\frac{4}{15}\right)\)\(e\left(\frac{3}{20}\right)\)\(e\left(\frac{43}{60}\right)\)\(e\left(\frac{2}{15}\right)\)\(e\left(\frac{11}{60}\right)\)\(e\left(\frac{17}{20}\right)\)\(e\left(\frac{7}{10}\right)\)\(e\left(\frac{29}{30}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{60})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 350 }(3,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{350}(3,\cdot)) = \sum_{r\in \Z/350\Z} \chi_{350}(3,r) e\left(\frac{r}{175}\right) = 10.3762132334+-8.2057418272i \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 350 }(3,·),\chi_{ 350 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{350}(3,\cdot),\chi_{350}(1,\cdot)) = \sum_{r\in \Z/350\Z} \chi_{350}(3,r) \chi_{350}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 350 }(3,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{350}(3,·)) = \sum_{r \in \Z/350\Z} \chi_{350}(3,r) e\left(\frac{1 r + 2 r^{-1}}{350}\right) = -2.1303625884+3.2804707101i \)