Properties

Conductor 864
Order 72
Real no
Primitive yes
Minimal yes
Parity even
Orbit label 864.bu

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(864)
 
sage: chi = H[13]
 
pari: [g,chi] = znchar(Mod(13,864))
 

Basic properties

sage: chi.conductor()
 
pari: znconreyconductor(g,chi)
 
Conductor = 864
sage: chi.multiplicative_order()
 
pari: charorder(g,chi)
 
Order = 72
Real = no
sage: chi.is_primitive()
 
pari: #znconreyconductor(g,chi)==1 \\ if not primitive returns [cond,factorization]
 
Primitive = yes
Minimal = yes
sage: chi.is_odd()
 
pari: zncharisodd(g,chi)
 
Parity = even
Orbit label = 864.bu
Orbit index = 47

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 ]
 

\(\chi_{864}(13,\cdot)\) \(\chi_{864}(61,\cdot)\) \(\chi_{864}(85,\cdot)\) \(\chi_{864}(133,\cdot)\) \(\chi_{864}(157,\cdot)\) \(\chi_{864}(205,\cdot)\) \(\chi_{864}(229,\cdot)\) \(\chi_{864}(277,\cdot)\) \(\chi_{864}(301,\cdot)\) \(\chi_{864}(349,\cdot)\) \(\chi_{864}(373,\cdot)\) \(\chi_{864}(421,\cdot)\) \(\chi_{864}(445,\cdot)\) \(\chi_{864}(493,\cdot)\) \(\chi_{864}(517,\cdot)\) \(\chi_{864}(565,\cdot)\) \(\chi_{864}(589,\cdot)\) \(\chi_{864}(637,\cdot)\) \(\chi_{864}(661,\cdot)\) \(\chi_{864}(709,\cdot)\) \(\chi_{864}(733,\cdot)\) \(\chi_{864}(781,\cdot)\) \(\chi_{864}(805,\cdot)\) \(\chi_{864}(853,\cdot)\)

Values on generators

\((703,325,353)\) → \((1,e\left(\frac{7}{8}\right),e\left(\frac{4}{9}\right))\)

Values

-11571113171923252931
\(1\)\(1\)\(e\left(\frac{7}{72}\right)\)\(e\left(\frac{31}{36}\right)\)\(e\left(\frac{11}{72}\right)\)\(e\left(\frac{49}{72}\right)\)\(e\left(\frac{1}{6}\right)\)\(e\left(\frac{11}{24}\right)\)\(e\left(\frac{5}{36}\right)\)\(e\left(\frac{7}{36}\right)\)\(e\left(\frac{5}{72}\right)\)\(e\left(\frac{8}{9}\right)\)
value at  e.g. 2

Related number fields

Field of values \(\Q(\zeta_{72})\)

Gauss sum

sage: chi.sage_character().gauss_sum(a)
 
pari: znchargauss(g,chi,a)
 
\( \tau_{ a }( \chi_{ 864 }(13,·) )\;\) at \(\;a = \) e.g. 2
\(\displaystyle \tau_{2}(\chi_{864}(13,\cdot)) = \sum_{r\in \Z/864\Z} \chi_{864}(13,r) e\left(\frac{r}{432}\right) = 0.0 \)

Jacobi sum

sage: chi.sage_character().jacobi_sum(n)
 
\( J(\chi_{ 864 }(13,·),\chi_{ 864 }(n,·)) \;\) for \( \; n = \) e.g. 1
\( \displaystyle J(\chi_{864}(13,\cdot),\chi_{864}(1,\cdot)) = \sum_{r\in \Z/864\Z} \chi_{864}(13,r) \chi_{864}(1,1-r) = 0 \)

Kloosterman sum

sage: chi.sage_character().kloosterman_sum(a,b)
 
\(K(a,b,\chi_{ 864 }(13,·)) \;\) at \(\; a,b = \) e.g. 1,2
\( \displaystyle K(1,2,\chi_{864}(13,·)) = \sum_{r \in \Z/864\Z} \chi_{864}(13,r) e\left(\frac{1 r + 2 r^{-1}}{864}\right) = -46.6000526724+-20.504513916i \)